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
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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 whatsoever can be drawn, e.g., ―The moon is for common sense. 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. Direct inference is the norm. Direct inference 3 Direct Logic is called ―direct‖ due to considerations such as the allows theories to directly speak of their own following: inference and to directly reason about Direct Logic does not incorporate general proof by relationships among the mutually inconsistent contradiction in a theory T. Instead it only allows self- code, specifications, and test cases of large refutation, e.g., (├ ) ├ (├ ). See T T T discussion below. 1 In Direct Logic, theories to speak directly about their If you have doubts, check out the kids and the VPs of major own inferability relation rather than having to resort to corporations. indirect propositions in a meta-theory.
2 Inference of Φ from Ψ in a theory T (Ψ ├ Φ) is Logicist and Logicism are used in this paper for the general T sense pertaining to logic rather than in the restricted technical ―direct‖ in the sense that it does not automatically sense of maintaining that mathematics is in some important sense incorporate the contrapositive i.e., it does not reducible to logic. automatically incorporate (Φ ├ Ψ). See discussion T below.
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Inconsistency is the Norm in Large Software down to any natural or given base; and when we cease our attempts to drive our piles into a deeper layer, it is not Systems because we have reached bedrock. We simply pause when “… find bugs faster than developers can fix them we are satisfied that they are firm enough to carry the and each fix leads to another bug” structure, at least for the time being. Or perhaps we do --Cusumano & Selby 1995, p. 40 something else more pressing. Under some piles there is no rock. Also some rock does not hold. The development of large software systems and the extreme dependence of our society on these systems have Different communities are responsible for constructing, introduced new phenomena. These systems have pervasive 4 evolving, justifying and maintaining documentation, use inconsistencies among and within the following: cases, and code for large, human-interaction, software Use cases that express how systems can be used systems. In specific cases any one consideration can trump and tested in practice5 the others. Sometimes debates over inconsistencies among Documentation that expresses over-arching the parts can become quite heated, e.g., between vendors. justification for systems and their technologies.6 In the long run, after difficult negotiations, in large Code that expresses implementations of systems software systems, use cases, documentation, and code all change to produce systems with new inconsistencies. Adapting a metaphor7 used by Karl Popper for science, the However, no one knows what they are or where they are bold structure of a large software system rises, as it were, located! above a swamp. It is like a building erected on piles. The piles are driven down from above into the swamp, but not Furthermore there is no evident way to divide up the code, documentation, and use cases into meaningful, consistent microtheories for human-computer interaction. 4 Moshe Vardi [2010] has defended the traditional paradigm of Organizations such as Microsoft, the US government, proving that program meet specifications and attacked an early and IBM have tens of thousands of employees pouring critical analysis as follows: ―With hindsight of 30 years, it seems over hundreds of millions of lines of documentation, that De Millo, Lipton, and Perlis' [1979] article has proven to be code, and use cases attempting to cope. In the course of rather misguided.‖ However, contrary to Vardi, limitations of the time almost all of this code will interoperate using Web traditional paradigm of proving that program meet specifications Services. A large software system is never done have become much more apparent in the last 30 years—as [Rosenberg 2007]. admitted even by some who had been the most prominent proponents, e.g., [Hoare 2003]. See discussion below. The thinking in almost all scientific and engineering work
5 has been that models (also called theories or microtheories) According to [Hoare 2009]: ―One thing I got spectacularly should be internally consistent, although they could be wrong. I could see that programs were getting larger, and I 8 thought that testing would be an increasingly ineffective way of inconsistent with each other. removing errors from them. I did not realize that the success of tests is that they test the programmer, not the program. Rigorous testing regimes rapidly persuade error-prone programmers (like me) to remove themselves from the profession. Failure in test 8 Indeed some researchers have even gone so far as to construct immediately punishes any lapse in programming concentration, consistency proofs for some small software systems, e.g., [Davis and (just as important) the failure count enables implementers to and Morgenstern 2005] in their system for deriving plausible resist management pressure for premature delivery of unreliable conclusions using classical logical inference for Multi-Agent code. The experience, judgment, and intuition of programmers Systems. In order to carry out the consistency proof of their who have survived the rigors of testing are what make programs system, Davis and Morgenstern make some simplifying of the present day useful, efficient, and (nearly) correct.‖ assumptions: No two agents can simultaneously make a choice 6 According to [Hoare 2009]: ―Verification [proving that (following [Reiter 2001] ). programs meet specifications] technology can only work against No two agents can simultaneously send each other errors that have been accurately specified, with as much inconsistent information. accuracy and attention to detail as all other aspects of the Each agent is individually serial, i.e., each agent can programming task. There will always be a limit at which the execute only one primitive action at a time. engineer judges that the cost of such specification is greater than There is a global clock time. the benefit that could be obtained from it; and that testing will be Agents use classical Speech Acts (see [Hewitt 2006b adequate for the purpose, and cheaper. Finally, verification 2007a, 2007c, 2008c]). [proving that programs meet specifications] cannot protect Knowledge is expressed in first-order logic. against errors in the specification itself.” The above assumptions are not particularly good ones for modern systems (e.g., using Web Services and many-core 7 Popper [1934] section 30. computer architectures). [Hewitt 2007a]
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Euclid [circa 300BC] in Alexandria, theories were Consistency has been the bedrock of mathematics intuitively supposed to be both consistent and complete. Wilhelm Leibniz, Giuseppe Peano, George Boole, When we risk no contradiction, Augustus De Morgan, Richard Dedekind, Gottlob Frege, It prompts the tongue to deal in fiction. Charles Peirce, David Hilbert, etc. developed mathematical Gay [1727] logic. However, a crisis occurred with the discovery of the logical paradoxes based on self-reference by Burali-Forti 9 Platonic Ideals were to be perfect, unchanging, and [1897], Cantor [1899], Russell [1903], etc. In response 10 eternal. Beginning with the Hellenistic mathematician Russell [1908] stratified types, [Zermelo 1905, Fränkel 1922, Skolem 1922] stratified sets and [Tarski and Vaught The following conclusions can be drawn for documentation, 1957] stratified logical theories to limit self-reference. use cases, and code of large software systems for human- Gödel [1931] and Rosser [1936] proved that nontrivial computer interaction: mathematical theories are incomplete, i.e., there are Consistency proofs are impossible for whole systems. propositions which can neither be proved nor disproved. There are some consistent subtheories but they are typically mathematical. There are some other consistent Consequently, although completeness and unrestricted self- microtheories as well, but they are small, make reference were discarded for general mathematics, the simplistic assumptions, and typically are inconsistent with other such microtheories [Addanki, Cremonini and bedrock of consistency remained. Penberthy 1989]. Nevertheless, the Davis and Morgenstern research programme to prove consistency of microtheories can be valuable for the Limitations of Classical Mathematical Logic theories to which it can be applied. Also some of the techniques As explained below, classical mathematical logic has that they have developed may be able to be used to prove the consistency of the mathematical fragment of Direct Logic and to limitations in that it is not inconsistency tolerant, has the prove inconsistency tolerance (see below in this paper). contrapositive inference bug, and cannot express direct argumentation. 9 ―The world that appears to our senses is in some way defective and filled with error, but there is a more real and perfect realm, Inconsistency Tolerance populated by entities [called ―ideals‖ or ―forms‖] that are eternal, Indeed, even at this stage, I predict a time when changeless, and in some sense paradigmatic for the structure and character of our world. Among the most important of these there will be mathematical investigations of [ideals] (as they are now called, because they are not located in calculi containing contradictions, and people space or time) are Goodness, Beauty, Equality, Bigness, Likeness, will actually be proud of having emancipated Unity, Being, Sameness, Difference, Change, and themselves from consistency. Changelessness. (These terms — “Goodness”, “Beauty”, and so Wittgenstein circa 1930 according to [Priest on — are often capitalized by those who write about Plato, in 2004] order to call attention to their exalted status;…) The most fundamental distinction in Plato's philosophy is between the many A limitation of classical logic for inconsistent theories is observable objects that appear beautiful (good, just, unified, that it supports the principle that from an inconsistency equal, big) and the one object that is what Beauty (Goodness, Justice, Unity) really is, from which those many beautiful (good, anything can be inferred, e.g. “The moon is made of green just, unified, equal, big) things receive their names and their cheese.” corresponding characteristics. Nearly every major work of Plato For convenience, I have given the above principle the is, in some way, devoted to or dependent on this distinction. Many name IGOR for Inconsistency in Garbage Out Redux.11 of them explore the ethical and practical consequences of IGOR can be formalized as follows in which a conceiving of reality in this bifurcated way. We are urged to contradiction about a proposition Ω infers any transform our values by taking to heart the greater reality of the proposition:12 [ideals] and the defectiveness of the corporeal world.” [Kraut 2004] Ω, ¬ Ω ├
10 Perfection has traditionally been sought in the realm of the spiritual. However, Kurtz and Ketcham [1993] expounded on the thesis of the ―spirituality of imperfection‖ building on the 11 In Latin, the principle is called ex falso quodlibet which means experience and insights of Hebrew prophets, Greek thinkers, that from falsity anything follows. Buddhist sages, Christian disciples and Alcoholics Anonymous. This is spirituality for the ―imperfect because it is real and 12 Using the symbol ├ to mean ―infers in classical mathematical because imperfect has the possibility to be real.‖ As Leonard Cohen said ―There is a crack in everything: that's how the light logic‖ and ⇨ to mean classical mathematical logical implication. gets in.‖ The conception that they present is very far from the Also ⇔ is used for logical equivalence, i.e., ―if and only if‖. Platonic Ideals of being perfect, unchanging, and eternal.
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The IGOR principle of classical logic may not seem very Contraposition also affects statistical probabilistic (fuzzy intuitive! So why is it included in classical logic? logic) systems as follows: Suppose (as above) Proof by contradiction: (├ , ) ⇨ (├ ) P(TrafficJam | Observe[WeekdayAt5PM]) = 118 which can be justified in classical logic on the P(TrafficJam) = 0 grounds that if Ψ infers a contradiction in a consistent then 퐏 퐎퐛퐬퐞퐫퐯퐞[퐖퐞퐞퐤퐝퐚퐲퐀퐭ퟓ퐏퐌] 퐓퐫퐚퐟퐟퐢퐜퐉퐚퐦 theory then Ψ must be false. In an inconsistent P(Observe[WeekdayAt5PM])= theory, proof by contradiction leads to explosion by 퐏 퐓퐫퐚퐟퐟퐢퐜퐉퐚퐦 퐎퐛퐬퐞퐫퐯퐞[퐖퐞퐞퐤퐝퐚퐲퐀퐭ퟓ퐏퐌] ) =019 the following derivation in classical logic by a which Thus contraposition is built into probabilistic (fuzzy logic) a contradiction about P infers any proposition : systems and consequently unwarranted inferences can be P, ¬P ├ ¬ ├ P, ¬P ├ (¬ ¬) ├ made.20 Disjunction introduction: (Ψ ├ (ΨΦ)) which in classical logic would say that if Ψ is true then Direct Argumentation 13 (ΨΦ)) is true regardless of whether Φ is true. In Integrity is when what you say, what an inconsistent theory, disjunction introduction leads you do, what you think, and who you to explosion via the following derivation in classical are all come from the same place. logic in which a contraction about P infers any Madelyn Griffith-haynie proposition : P,¬ P ├ (P),¬ P ├ In the Tarskian framework of classical mathematical logic, a theory cannot directly express argumentation.21 For example a classical theory cannot directly represent its own Contrapositive Inference Bug inference relationship and consequently cannot directly You can use all the quantitative data you represent its rules of inference. This kind of restriction has can get, but you still have to distrust it and been challenged by [Feferman 1984a]: use your own intelligence and judgment. …natural language abounds with directly or Alvin Toffler indirectly self-referential yet apparently harmless expressions—all of which are excluded from the Direct inference is used in to directly infer conclusions Tarskian framework. from premises. For example, suppose that we have Direct Argumentation means that ├ in a proposition
A1. Observe[WeekdayAt5PM] ├ TrafficJam14 22 means inference. For example, ├ and ├ infer
A2. ├ TrafficJam15
18 In classical logic, Observe[WeekdayAt5PM] is inferred from The probability is 1 for TrafficJam given A1 and A2 above.16 Consequently, the contrapositive Observe[WeekdayAt5PM]. inference bug comes into play even in the absence of overt 19 17 Varying P(TrafficJam) doesn't change the principle involved inconsistency. because P(Observe[WeekdayAt5PM]) ≦ P(TrafficJam)
20 This example illustrates that the choice of how to incorporate 13 Disjunction introduction follows from the principle of measurements into statistics can effectively determine the model absorption that Ψ is equivalent to Ψ(ΨΦ). Consequently, in being used. In this particular case, the way that measurements Direct Logic absorption must be limited to inferring Ψ from were taken did not happen to take into account things like Ψ(ΨΦ). holidays and severe snow storms. This point was largely missed in [Anderson 2008]. which stated 14 Observing weekday at 5PM infers a traffic jam “Correlation is enough.” We can stop looking for
15 models. We can analyze the data without hypotheses No traffic jam about what it might show. We can throw the numbers
16 into the biggest computing clusters the world has ever Note that contrapositive for inference should not be confused seen and let statistical algorithms find patterns where with contrapositive for implication because A1 is different from science cannot. (emphasis added) the following:├ Observe[WeekdayAt5PM] ⇨ TrafficJam 21 where ⇨ is implication. See discussion below. Instead resort is usually made to meta-theory stratification, reification or provability logic. See below. 17 Many in the nonmonotonic community have not included 22 Cf. ―on the ordinary notion of proof, it is compelling just contraposition in rules, e.g., [Reiter 1980; Prakken and Sartor because, presented with it, we cannot resist the passage 1996; Caminda 2008]. According to [Ginsberg 1994 pg. 16]: ―although almost all of the symbolic approaches to nonmonotonic from premises to conclusion without being unfaithful to the reasoning do allow for the strengthening of the antecedents of meanings we have already given to the expressions default rules, many of them do not sanction contraposition of employed in it.‖ [Dummett 1973] these rules.‖
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├ , which in Direct Logic can be expressed as follows Provide a foundation for reasoning about the by Direct Argumentation: mutually inconsistent implementation, specifications, and test cases large software systems. , (├ ) ├ Formalize a notion of “direct” inference for
The above proposition fulfills the demand of the Tortoise reasoning about inconsistent information that Support all “natural” deductive inference [Fitch Whatever Logic is good enough to tell me is worth 1952; Gentzen 1935] with the exception of general writing down. Proof by Contradiction and Disjunction 25 in the famous paradox in ―What the Tortoise said to Introduction. Achilles.‖ Carroll [1895] Support all the usual Boolean equivalences among propositions with exception of absorption, which must be restricted one way to avoid IGOR. Direct Logic Support roundtripping among code, documentation, The proof of the pudding is the eating. and use cases of large software systems. (See Cervantes [1605] in Don Quixote. Part 2. Chap. 24 discussion below.) Provide increased safety in reasoning using inconsistent information. Direct Logic is a minimal fix to make
mathematical logic and statistical probability Direct Logic supports direct inference26 (├ ) for an (fuzzy) inference adequate for the T Cervantes [1605] in Don Quixote. Part 2. Chap. 24 inconsistent theory . Consequently, ├ does not support requirements of large-scale Internet T T applications (including sense making for either general proof by contradiction or disjunction
natural language). introduction. However, ├ does support all other rules of T 27 Direct Logic23 is an inconsistency tolerant24 inference natural deduction [Fitch 1952; Gentzen 1935]. Consequently, Direct Logic is well suited for practical system for reasoning about large software systems with the 28 following goals: reasoning about large software systems. The theories of Direct Logic are ―open‖ in the sense of open-
23 ended schematic axiomatic systems [Feferman 2007b]. The Direct Logic is distinct from the Direct Predicate Calculus language of a theory can include any vocabulary in which its [Ketonen and Weyhrauch 1984]. axioms may be applied, i.e., it is not restricted to a specific
vocabulary fixed in advance (or at any other time). Indeed a 24 Paraconsistent logic (name coined by Francisco Miró Quesada in 1976 [Priest 2002, pg. 288]) is a special form of inconsistency tolerant logic in which an inconsistency does not infer every Paraconsistent logics have not been satisfactory for the purposes proposition. Unfortunately, paraconsistency is a very weak of Software Engineering because of their many seemingly property. For example, adding the following rule: arbitrary variants and their idiosyncratic inference rules and Ψ, ¬Ψ ├ GreenCheese[Moon] notation. For example (according to Priest [2006]), most preserves paraconsistency because not all propositions are paraconsistent and relevance logics rule out Disjunctive inferred from an inconsistency. Syllogism ((ΦΨ), ¬Φ ├ Ψ). However, Disjunctive Syllogism The most extreme form of paraconsistent mathematics is seems entirely natural for use in Software Engineering! dialetheism [Priest and Routley 1989] which maintains that there are true inconsistencies in mathematics itself e.g., the Liar 25 In this respect, Direct Logic differs from Quasi-Classical Logic Paradox. However, mathematicians (starting with Euclid) have [Besnard and Hunter 1995] for applications in information worked very hard to make their theories consistent and systems, which does include Disjunction Introduction. inconsistencies have not been an issue for most working mathematicians. As a result: 26 Direct inference is defined differently in this paper from Since inconsistency was not an issue, mathematical logic probability theory [Levy 1977, Kyburg and Teng 2001], which focused on the issue of truth and a model theory of truth refers to ―direct inference‖ of frequency in a reference class (the was developed [Dedekind 1888, Löwenheim 1915, most specific class with suitable frequency knowledge) from Skolem 1920, Gödel 1930, Tarski and Vaught 1957, which other probabilities are derived. Hodges 2006]. More recently there has been work on the
development of an unstratified logic of truth [Leitgeb 27 2007, Feferman 2007a]. But with the modification from classical natural deduction that Paraconsistent logic somewhat languished for lack of Ψ├ Φ does not automatically mean that ├ (Ψ⇨Φ). See T T subject matter. The lack of subject matter resulted in discussion of two-way deduction theorem below. paraconsistent systems that were for the most part so awkward as to be unused for mathematical practice. 28 In this respect, Direct Logic differs from previous Consequently mainstream logicians and mathematicians have inconsistency tolerant logics, which had inference rules that made tended to shy away from paraconsistency. them intractable for use with large software systems.
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theory can be an open system can receive new information at (implication), and ⇔ (bi-implication) are any time [Hewitt 1991, Cellucci 1992]. propositions. Atomic names are expressions.32 Also numbers are In the argumentation lies the knowledge expressions. Testimony is like an arrow shot from a If x1, …, and xn are variables and is a proposition, long-bow; its force depends on the then the following is a proposition that says ―for all x1, strength of the hand that draws it. But …, and xn: holds: argument is like an arrow from a cross- x1; …; xn : bow, which has equal force if drawn by a If F is an expression and E1, …, En are expressions, child or a man. Charles Boyle then F(E , …, E ) is an expression. 1 n Partly in reaction to Popper29, Lakatos [1967, §2]) calls the If X1, …, Xn are identifiers and E is an expression, view below Euclidean:30 then (λ(X1, …, Xn) E) is an expression. “Classical epistemology has for two thousand years If E1, E2, and E3 are expressions, then the following modeled its ideal of a theory, whether scientific or are expressions: mathematical, on its conception of Euclidean if E1 then E2 else E3 geometry. The ideal theory is a deductive system with E1 = = E2 (E1 and E2 are the same Actor) an indubitable truth-injection at the top (a finite If E1, …, En are expressions, then [E1, …, En] (the conjunction of axioms)—so that truth, flowing down sequence of E1, …, and En) is an expression from the top through the safe truth-preserving If E1 and E2 are expressions, [E1 E2] (the sequence channels of valid inferences, inundates the whole of E1 followed by the elements of the sequence E2) is system.” an expression If X is a variable, E is an expression, and is a Since truth is out the window for inconsistent theories, we proposition, then {XE | } (the set of all X in E such have the following reformulation: that ) is an expression. Inference in a theory T (├ ) carries argument If E1 and E2 are expressions, then E1=E2, E1E2 and T from antecedents to consequents in chains of E1E2 are propositions inference. If P is an expression and E1, …, En are expressions, then P[E1, …, En] is a proposition. Syntax of Direct Logic If E1 and E2 are expressions, then E1 E2 (E1 can reduce to E in the nondeterministic λ-calculus) is a The aims of logic should be the 2 proposition. creation of ―a unified conceptual If E is an expression, then E (E always converges in apparatus which would supply a the nondeterministic λ-calculus) is a proposition. common basis for the whole of human If E is an expression, then E (E is irreducible in the knowledge.” [Tarski 1940] nondeterministic λ-calculus) is a proposition. The syntax of Direct Logic is defined by expressions31 and If E1 and E2 are expressions, then E1E2 (E1 can propositions as defined below: converge to E2 in the nondeterministic λ-calculus) is a proposition. If and are propositions then, (negation), If E is an expression, then E (E reduces to exactly 1 (conjunction), (disjunction), ⇨ 1 expression in the nondeterministic λ-calculus) is a proposition. If T is an expression and is a proposition, then 29 Proof by contradiction has played an important role in science ⊢ ( can be inferred in ) is a proposition. ⊬ is (emphasized by Karl Popper [1962]) as formulated in his T T principle of refutation which in its most stark form is as follows: the negation of ⊢. If ├ Ob for some observation Ob, then it can be T concluded that T is refuted (in a theory called Popper ), i.e., ├ Popper T See Suppe [1977] for further discussion.
30 although there is no claim concerning Euclid‘s own orientation 32 For example, Fred and x are atomic names. An atomic name is a constant, variable or identifier. Variables are universally 31 Notionally, expressions take their values from an extension to quantified and identifiers are bound in λ- expressions. As a XML (see appendix). convention in this paper, the first letter of a constant will be capitalized.
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If T is an expression and 1, …, k are propositions Transitivity is perhaps the most fundamental principle of inference.36 and 1, …, n are propositions then
37 1, …, k ⊢ 1, …, n is a proposition that says T Transitivity of Argumentation: ( , … and ) infer ( , …, and ) in 1 k 1 n T. (├ ), (├ ) ├ (├ ) If is an expression, E is an expression and is a T T T T T argumentation is transitive proposition, then E ╟ (E is an argument for in ) T T is a proposition. In Direct Logic, transitivity gives rise to Detachment (a If s is a sentence (in XML33). then s (the version of modus ponens for inference) as follows: abstraction of s) is a proposition. If p is a phrase (in , (├ ) ├ T T XML), then 34 McGee [1985] has challenged modus ponens using an p (the abstraction of p) is an expression. example that can be most simply formalized in Direct If is a proposition, then (the reification of ) is Logic as follows:38 a sentence (in XML). If E is an expression, then RepublicanWillWin├ (ReaganWillWin├ AndersonWillWin)39 E (the reification of E) is a phrase (in XML). McGee McGee ├ RepublicanWillWin40 McGee In general, the theories of Direct Logic are inconsistent and therefore propositions cannot be consistently labeled with From the above, in Direct Logic it follows that: truth values. ReaganWillWin ├ AndersonWillWin Note that Direct Logic does not have quantifiers, but McGee universally quantified variables are allowed in top level 35 McGee challenged the reasonableness of the above propositions called statements. conclusion on the grounds that. intuitively, the proper inference is that if Reagan will not win, then Transitivity of Argumentation AndersonWillWin because Carter (the Democratic Scientist and engineers speak in the name of candidate) will win. However, in theory McGee, it is new allies that they have shaped and enrolled; reasonable to infer AndersonWillWin from ReaganWillWin representatives among other representatives, because RepublicanWillWin holds in McGee. they add these unexpected resources to tip the balance of force in their favor. Latour [1987] Second Principle
33 Computer science has standardized on XML for the (textual) representation of tree structures.
34 For example, λ(x) x=0 is an expression. In this respect
Direct Logic differs from Lambda Logic [Beeson 2004], which 36 does not have abstraction and reification. [Patel-Schneider 1985] developed a logical system without transitivity in order to make inference recursively decidable. 35 Consider following statement S: 37 The fundamental principles of Direct Logic are placed in boxes p,qHumans: Mortal[ACommonAncestor(p, q)] like this one and they are not independent. where the syntax has been extended in the obvious way to allow constraints on variables. 38 McGee phrased his argument in terms of implication which in An instantiation of S can be specified by supplying values for Direct Logic (see following discussion in this paper) would be as follows: variables. For example S[Socrates, Plato] is the proposition ├ RepublicanWillWin⇨ (ReaganWillWin⇨AndersonWillWin) Socrates,PlatoHumans⇨ McGee ├ RepublicanWillWin Mortal[ACommonAncestor(Socrates, Plato)] McGee Note that care must be taken in forming the negation of However, this makes no essential difference because, in Direct statements. Logic, it still follows that Direct Logic directly incorporates Skolemization unlike Lambda ├ (ReaganWillWin ⇨ AndersonWillWin) McGee Logic [Beeson 2004], classical first-order set theory, etc. For 39 example the negation of S is the proposition In the election, there were only two Republican candidates: Reagan and Anderson. Consequently, in theory McGee ,from (PS,QSHumans ⇨ Mortal[ACommonAncestor(PS QS)]) RepublicanWillWin, it follows that AndersonWillWin can be where PS and QS are Skolem constants. See the axiomatization of inferred from ReaganWillWin. set theory in the first appendix for further examples of the use of 40 Skolem functions in Direct Logic. RepublicanWillWin holds in theory McGee .
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Reiteration: ├ a proposition infers itself Soundness, Faithfulness, and Adequacy T Exchange: (,├ ) ┤├ (,├ ) Soundness in Direct Logic is the principle that the rules of T T T (├ ,) ┤├ (├ ,) Direct Logic preserve arguments, i.e., T T T the order of propositions are written does not matter
Residuation: (,├ ) ┤├ (├ (├ )) Soundness of Argumentation: T T T T hypotheses may be freely introduced and discharged (├ ) ├ ((├ ) ├ (├ )) Monotonicity: (├ ) ├ (, ├ ) T T T T T T T T when an argument holds and furthermore the an argument remains if new information is added antecedent of the argument holds, infer that the Dropping: (├ ,) ├ (├ ) T T T consequence of the argument holds. an argument remains if extra conclusions are dropped Adequacy is the property that a proposition holding a Argument combination:((├ ), (├ )) ┤├ (├ ,) T T T T theory infers that there is an argument for the proposition arguments can be combined in the theory. i.e., Variable Elimination: (x: P[x]) ├ P[E] T Adequacy of Argumentation: a universally quantified variable of a statement can be instantiated with any expression E (taking care that none of (├ ) ├ (├ (├ )) the variables in E are captured). T T T T when an inference holds, infer that there is an Variable Introduction: Let Z be a new constant argument that the inference holds (├ P[Z] ) ├ (├ x: P[x]) T T T inferring a statement with a universally quantified Faithfulness is the property that when a theory holds that variable is equivalent to inferring the statement with a newly introduced constant substituted for the variable an argument holds in the theory, then the theory faithfully holds the argument, i.e., Self-refutation Faithfulness of Argumentation: “Contrariwise,” continued Tweedledee, “if it was so, (├ (├ )) ├ (├ ) it might be; and if it were so, it would be; but as it T T T T When it holds that an argument holds, infer that isn't, it ain't. That's logic.” Carroll [1871] the argument holds. Direct Logic supports self-refutation, which is a restricted Housekeeping version of proof by contradiction as follows:
Logic merely sanctions the conquests of the intuition. Self Infers Opposite: Jacques Hadamard (quoted in Kline [1972]) (├ ) ├ (├ ) T T T Direct Logic has the following housekeeping rules:41 when a proposition infers its own negation, its negation holds Self Infers Argument for Opposite: (├ (├ )) ├ (├ ) when a TpropositionT infers Tits own Tnegation holds, its negation holds Argument for Self Infers Opposite: ((├ )├ ) ├ (⊬ ) when Tthat a propositionT holdsT infersT the negation of the proposition, the proposition does not hold Argument for Self Infers Argument for Opposite: ((├ )├ (├ )) ├ (⊬ ) T T T T T when that a proposition holds infers the negation 41 Nontriviality principles have also been proposed as extensions of the proposition holds, the proposition does to Direct Logic including the following: not hold
Direct Nontriviality: () ├ (├ ) T T Self-refutation can sometimes do inferences that are the negation of a proposition infers that it cannot be 42 inferred traditionally done using proof by contradiction. For Meta Nontriviality: (├ ) ├ (├ ) T T T the inference of the negation of a proposition infers that the proposition cannot be inferred. 42 Provability logic [Gödel 1931; Hilbert and Bernays 1939; Löb 1955; Verbrugge 2003] is an approach for expressing some
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example the proof of the incompleteness of theories in this Direct Logic defines disjunction () in terms of paper makes use of Self-refutation. conjunction and negation in a fairly natural way as follows:
Booleans Disjunction in terms of The Booleans43 in Direct Logic are as close to classical Conjunction and Negation: logic as possible. ┤├ () T Negation We could not even think „being‟ without Since Direct Logic aims to preserve standard Boolean a double negation: being must be properties, we have the following principle: thought as the negation of negation of being. Conjunction infers Disjunction: Paul Tillich [2000] () ├ () The following is a fundamental principle of Direct Logic:44 T Double Negation Elimination: Implication ┤├ 45 T Lakatos characterizes his own view as quasi-empirical: “Whether a deductive system is Euclidean or quasi- Other fundamental principles for negation are found in the empirical is decided by the pattern of truth value flow next sections. in the system. The system is Euclidean if the characteristic flow is the transmission of truth from Conjunction and Disjunction the set of axioms „downwards‟ to the rest of the Direct Logic tries to be as close to classical logic as system—logic here is an organon of proof; it is possible in making use of natural inference, e.g., ―natural quasi-empirical if the characteristic flow is deduction‖. Consequently, we have the following retransmission of falsity from the false basic equivalences for juxtaposition (comma): statements „upwards‟ towards the „hypothesis‟—logic here is an organon of criticism.” Conjunction in terms of Juxtaposition (comma): Direct Logic defines implication (⇨) in terms of (, ├ ) ┤├ ()├ conjunction and negation in a fairly natural way as follows: T T T (├ , ) ┤├ ├ () T T T Implication in terms of Conjunction and Negation: ⇨ ┤├ ( ) T argumentation in the Tarskian framework. Provability logic differs from Direct Logic in many ways including the following: Two-way Deduction Theorem Provability logic is not inconsistency tolerant In classical logic there is a strong connection between Provability logic cannot express its own provability deduction and implication through the Classical Deduction relation ├ Theorem: ProvablityLogic Albert Visser [private communication] has kindly pointed out (├(Ψ⇨Φ)) ┫├ (Ψ├ Φ) that all the principles of self-refutation (except Self Infers Opposite) are not valid in classical Provability Logic when However, the classical deduction theorem does not hold in 46 (├ ) is interpreted as Prov( ⇨ ). general for theories of Direct Logic. Instead, Direct Logic has a Two-way Deduction Theorem that is explained 43 (negation), (conjunction), (disjunction), and below. (implication).
44 Intuitionistic logic has a limited form of this principle as follows: ┤├ . Unfortunately, 46 Intuitionism For example, in the empty theory ⊥ (that has no axioms Intuitionism has IGOR: , ¬ ├ .
Intuitionism beyond those of Direct Logic),Q├ (P P) but ⊬ (Q ⇨ ⊥ ⊥ (P P)). 45 ┤├ means ├ and ├ T T T
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Lemma (├ (⇨)) ├ ((├ ) (├ )) T T T T Boolean equivalences Proof: Suppose ├ (⇨) T Theorem: The following usual Boolean equivalences hold: Therefore ├ () T By Disjunctive Syllogism, it follows that ├ and T Double Negation: ┤├ ├ . T T Idempotence of : ┤├ What about the converse of the above theorem? T Commutativity of : ┤├ Lemma ((├ ) (├ )) ├ (├ (⇨)) T T T T T Proof: Suppose ├ and ├ Associativity of : () ┤├ () T T T By the principle of Simple Self-refutation, to prove Distributivity of over : ├ (⇨), it is sufficient to prove the following: T () ┤├ () () (⇨)├ (⇨) T T De Morgan for : () ┤├ Thus it is sufficient to prove ()├ () T T But ()├ () ├ () by the Idempotence of : ┤├ T T T suppositions above and the principle that Conjunction Commutativity of : ┤├ Infers Disjunction. T Associativity of : () ┤├ () T Putting the above two theorems together we have the Two- Distributivity of over : Way Deduction Theorem for Implication: () ┤├ () () (├ (⇨)) ┫├ ((├ ) (├ )) T T T T T De Morgan for : () ┤├ Consequently: T Implication as Disjunction: ⇨ ┤├ In Direct Logic, implication carries argument T both ways between antecedents and consequents. Contrapositive: ⇨ ┤├ ⇨ Thus, in Direct Logic, implication (⇨), rather than T inference (├ ), supports Lakatos quasi-empiricism. T Splitting 48 ―Taking the principle of excluded middle from The following corollaries follow: the mathematician would be the same, say, as * Two-Way Deduction Theorem for Disjunction:47 proscribing the telescope to the astronomer …” ├ ()) ┫├ ((├ ) (├ )) T T T T Hilbert [1920] * Transitivity of Implication: (⇨), (⇨) ├ (⇨) Direct Logic has the principle of Splitting that can be T expressed as follows: Proof: Follows immediately from the Two-Way Deduction Theorem for Implication by chaining Splitting: in both directions for . (), (├ ), (├ ) ├ * Reflexivity of Implication: ├ (⇨) T T T T Proof: Follows immediately from ├ and T Splitting is very useful in reasoning by cases. ├ using the Two-Way Deduction T Theorem. Splitting by Negation: ├ () Theorem. T Proof: Follows immediately from Reflexivity of Implication, the definition of implication, De Morgan, and Double Negation Elimination.
48 47 which explains that ⊬ () in general because ―Excluded middle‖ is the traditional name for Splitting by T Negation (see below). ⊬ ( ⊢ ( ⊢ )) T T T
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Absorption For example: Direct Logic supports the part of Absorption that does not lead to IGOR: ⊬⊥ ⇨ (⇨) ├⊥ (├⊥)
⊬⊥ (⇨ ) ⇨ (⇨ (⇨ )) Absorption of : () ├ (⇨ ) ├⊥ (⇨ (⇨ )) T Absorption of : () ├ ⊬⊥ ⇨ () ├⊥ () T ⊬⊥( ( )) ⇨ ( ) ├⊥ Decidability of Inference in Boolean Direct Logic All “philosophically interesting” ⊬⊥ ( ( )) ⇨ () ├⊥ propositional49 calculi for which the decision problem has been solved ⊬⊥ ( ()) ⇨ () ├⊥ have been found to be decidable Harrop [1965] Inconsistency tolerance facilitates theory Boolean Direct Logic is an important special case in which development the propositions are restricted to being composed of atomic proposition50 connected by negation, conjunction, A little inaccuracy sometimes disjunction, and implication. saves tons of explanation. Saki in ―The Square Egg‖ In Boolean Direct Logic, to decide whether 1├⊥ 2, 51 first put 1 and 2 in conjunctive normal form and apply Inconsistency tolerant theories can be easier to develop following transformation on 2: than classical theories because perfect absence of inconsistency is not required. In case of inconsistency, ├⊥ 1, 2 (├⊥ 1) (├⊥ 2)
there will be some propositions that can be both proved and disproved, i.e., there will be arguments both for and For each conjunct 12 in 1 use the following transformation: against the propositions. A classic case of inconsistency occurs in the novel , (12)├⊥ ,(1 ├⊥2),(2 ├⊥1) ├⊥ Catch-22 [Heller 1995] which states that a person ―would be crazy to fly more missions and sane if he didn't, but if he For each conjunct 12 in 2 use the following was sane he had to fly them. If he flew them he was crazy transformation: and didn't have to; but if he didn't want to he was sane and had to. Yossarian was moved very deeply by the absolute ├⊥(12) (,1├⊥ 2)(,2├⊥ 1) simplicity of this clause of Catch-22 and let out a Thus the decision problem for inferences in Boolean respectful whistle. „That's some catch, that Catch-22,‟ he Direct Logic reduces to deciding problems of the form observed.‖
├⊥ where is a list consisting of literals and literal inferences52 and is a literal. These subproblems are decided using splitting on ├⊥ () for each literal using:
├⊥ (, ├⊥ ) (, ├⊥ )
49 i.e., having only Boolean connectives
50 An atomic proposition is just an identifier like P or Q.
51 A proposition is in conjunctive normal form when it is the conjunction of clauses, where each clause is a disjunction of literals
52 A literal inference is a proposition of the form ├⊥ , where is a list of literals and is a literal.
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In the spirit of Catch-22, consider the follow Reification and Abstraction axiomization of the above: Direct Logic distinguishes between concrete sentences in XML and abstract propositions.54 Software Engineering 1. p: AbleToFly[p], Fly[p] ├ Sane[p] Catch-22 requires that it must be easy to construct abstract axiom propositions from concrete sentences.55 Direct Logic 2. p: Sane[p] ├ Obligated[p, Fly] axiom provides abstraction for this purpose as follows: Catch-22 Every sentence s in XML has an abstraction56 3. p: Sane[p],ObligatedToFly[p]├ Fly[p] 57 Catch-22 that is the proposition given by s. axiom 4. ├ AbleToFly[Yossarian] axiom s,tSentences: s=t ⇨ ( s ⇔ t ) Catch-22 5. Fly[Yossarian] ├ Fly[Yossarian] Catch-22 from 1 through 4 Abstraction can be used to formally self-express important properties of Direct Logic such as the following: 6. ├ Fly[Yossarian] Catch-22 from 5 using Self Infers Opposite The principle Inferences have Arguments 7. p: Fly[p] ├ Crazy[p] axiom Catch-22 says that is inferred in theory T if and only 8. p: Crazy[p] ├ ObligatedToFly[p] axiom Catch-22 if has a argument that establishes it in T, 9. p: Sane[p], ObligatedToFly[p] ├ Fly[p] Catch-22 i.e. ╟ axiom T 10. ├ Sane[Yossarian] axiom s,tSentences: ├ s AnArgument (s) ╟ s Catch-22 T T T where AnArgument is a choice function that 11. ├ Fly[Yossarian] from 6 through 10 T Catch-22 chooses an argument for s 58 Thus there is an inconsistency in the above theory Furthermore, there is a linear recursive ArgumentChecker such that: Catch-22 in that: T 6. ├ Fly[Yossarian] (xArguments; sSentences: Catch-22 11. ├ Fly[Yossarian] ArgumentChecker (x, s)=1 ⇔ x ╟ s) Catch-22 T T
Various objections can be made against the above Conversely, every proposition has a reification59 (given 53 axiomization of the theory Catch-22. However, by 60) that is a sentence in XML.61 Catch-22 illustrates several important points:
54 Even a very simple microtheory can engender This is reminiscent of the Platonic divide (but without the inconsistency moralizing). Gödel thought that ―Classes and concepts may, Inconsistency tolerance facilitates theory however, also be conceived as real objects…existing independently of our definitions and constructions.‖ [Gödel 1944 development because a single inconsistency is pg 456] not disastrous. Direct Logic supports fine grained reasoning 55 Analogous the requirement that it must be easy to construct because inference does not necessarily carry executable code from concrete programs (text). argument in the contrapositive direction. For example, the general principle ―A person who 56 For example, if s and t are sentences in XML, then flies is crazy.‖ (i.e., Fly[p] ├ Crazy[p] ) does Catch-22 ├
53 Both Crazy[Yossarian] and Sane[Yossarian] can be inferred 59 Reifications are in some ways analogous to Gödel numbers from the axiomatization, but this per se is not inconsistent. [Gödel 1931].
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Direct Logic makes use of the following: The sections below address issues concerning the use of Logical Roundtripping Principle:63 abstraction and reification. If is Admissible for T then: Roundtripping ├ ( ⇔ ) T To thine own self be true. And it must follow, as the night the day, Of course, the above criterion begs the questions of which Thou canst not then be false to any man. propositions are Admissible in T! A proposed answer is Shakespeare in-―Hamlet‖ Act 1, scene iii. provided by the following:
Roundtripping is the process of going back and forth using 62 The Criterion of Admissibility for Direct abstraction and reification. Roundtripping is becoming Logic is64: increasingly important in software engineering. e.g., The execution of code can be dynamically checked is Admissible for T if and only if against its documentation. Also Web Services can () ├ (├ ) be dynamically searched for and invoked on the T T I.e., the Criterion of Admissibility is that a basis of their documentation. Use cases can be inferred by specialization of proposition is Admissible for a theory T if and only documentation and from code by automatic test if its negation infers in T that its negation can be generators and by model checking. inferred in T Code can be generated by inference from documentation and by generalization from use In other words a proposition is Admissible when its cases. negation infers that there is an argument for its negation holding.65 Abstraction and reification are needed for large software systems so that that documentation, use 63 cases, and code can mutually speak about what has Logical roundtripping says if is Admissible for T then its been said and its meaning. reification has enough information that abstracting back is logically equivalent to in T. Roundtripping for the nondeterministic λ-calculus is discussed in the appendix.
64 Note that there is an asymmetry in the definition of Admissibility with respect to negation. In general, it does not follow that is admissible for T just because is admissible 60 Heuristic: Think of the ―elevator bars‖ . . . around as for T. The asymmetry in Admissibility is analogous to the ―lowering‖ the abstract proposition ―down‖ into the concrete asymmetry in the Criterion of Refutability [Popper 1962]. For sentence that is its reification in XML. example the sentence “There are no green swans.” is readily The reifications of a propositions can be quite complex because refuted by the observation of a green swan. However, the of various optimizations that are used in the implementations of negation is not so readily refuted. propositions. Also note that admissibility is different from the following: ├ ( ⇨ ├ ) T T 61 Note that, if s is a sentence, then in general s s. which is equivalent to the following: ├ ((├ ) ⇨ ) T T 62 To avoid inconsistencies in mathematics (e.g., Liar Paradox, The above statement illustrates a problem with the traditional Russell‟s Paradox, Curry‟s Paradox, etc.), some restrictions are concept of “Negation as Failure” that was first noted in needed around logical roundtripping. The question is how to do it connection with the development of Planner, namely, “The [Feferman 1984a, Restall 2006].62 dumber the system, the more it can prove!” See the discussion The approach which is currently standard in mathematics is the on the limitations of Logic Programming. Tarskian framework of assuming that there is a hierarchy of 65 metatheories in which the semantics of each theory is formalized Admissibility is a generalization of the property of being in its metatheory [Tarski and Vaught 1957]. GoldbachLike (emphasized by [Franz 2005]) which is defined to Large software systems likewise abound with roundtripping in be all sentences s of arithmetic (ℕ) such that reasoning about their use cases, documentation, and code that are fExpressions s=nω f(n) BoundedQuantification(f) excluded by the Tarskian framework. Consequently the where BoundedQuantification(f) means that all the assumption of hierarchical metatheories is not very suitable for quantifiers in f are bounded, i.e., all quantifiers are of one of Software Engineering. the following two forms: 1. variableexpression…
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.f(1)1 f(1)2 f(1)3 … f(1)i …
Theorem. If and are Admissible for T, then is .f(2)1 f(2)2 f(2)3 … f(2)i …
Admissible for T. .f(3)1 f(3)2 f(3)3 … f(3)i … Proof. Suppose and are Admissible for T, i.e., … () ├ (├ ) and () ├ (├ ). The goal is .f(i)1 f(i)2 f(i)3 … f(i)i … T T T T … to prove () ├ (├ ()), which is T T equivalent to () ├ (├ ()), which T T Define Diagonal as follows: follows immediately from the hypothesis. Diagonal ≡ Diagonalize(f) Theorem. If and are Admissible for T, then ⇨ 66 where Diagonalize(g) ≡ λ(i) g(i)i is Admissible for T. where g(i) is the complement of g(i) Proof. (⇨) ( ). Therefore the theorem i i follows from the previous theorem by Double Negation Diagonal can be diagrammed as follows: Elimination. .f(1) f(1) f(1) … f(1) … The motivation for Admissibility builds on the 1 2 3 i denotational semantics of the Actor model of computation .f(2)1 f(2)2 f(2)3 … f(2)i … which were first developed in [Clinger 1981]. .f(3)1 f(3)2 f(3)3 … f(3)i … Subsequently [Hewitt 2006] developed the …
TimedDiagrams model with the Computational .f(i)1 f(i)2 f(i)3 … f(i)i … Representation Theorem that is discussed later in this … paper. Therefore Diagonal is a real number not enumerated In this context, is Admissible for S means that by f because it differs in the ith digit of every f(i). implies that there is a counter example to in DenoteS so The Diagonal Argument is used in conjunction with the that in the denotational theory S induced by the system S: Logical Fixed Point theorem that is described in the next section. () ├S (├S)
Logical Fixed Point Theorem Theorem. For every which is Admissible for T, there is The Logical Fixed Point Theorem enables propositions to an argument such that: effectively speak of themselves . In this paper, the fixed point theorem is used to ├ ArgumentChecker ( , )=1 T T demonstrate the existence of self-referential sentences that However, using logical roundtripping can result in will be used to prove theorems about Direct Logic using paradoxes as a result of the Diagonal Argument (explained the Diagonal Argument. below). Theorem [a λ-calculus version of Carnap 1934 pg 91 after Diagonal Argument Gödel 1931]67: The Diagonal Argument [du Bois-Reymond 1880] has Let f be a total function from Sentences to been used to prove many famous theorems beginning with Sentences68 the proof that the real numbers are not countable [Cantor ├ ( Fix(f) ⇔ f(Fix(f)) ) 1890, Zermelo 1908]. T where Fix(f) ≡ Θ(Θ) Proof. Suppose to the contrary that the function which exists because f always converges f:ℕℝ enumerates the real numbers that are where Θ ≡ λ(g) f(λ(x) (g(g))(x))69 greater than equal to 0 but less than 1 so that f(n)i is the ith binary digit in the binary expansion of 66 f(n) which can be diagrammed as an array with The symbol ―≡‖ is used for ―is defined as‖. infinitely many rows and columns of binary digits 67 as follows: Credited in Kurt Gödel, Collected Works vol. I, p. 363, ftn. 23. However, Carnap, Gödel and followers did not use the λ calculus and consequently their formulation is more convoluted.
2. variable expression… 68 Note that f is an ordinary Lisp-like function except that where variable does not appear in expression Sentences (a subset of XML) are used instead of S-expressions. Theorem. If is Goldbach-like, then is Admissible for ℕ.
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Proof However LiarProposition is not admissible for ⊥ Fix(f) = Θ(Θ) = λ(g) f(λ(x) (g(g))(x)) (Θ) because presumably = f (λ(x) (Θ(Θ))(x)) LiarProposition ⊬⊥ (⊢⊥ LiarProposition) = f (Θ(Θ)) Likewise other standard paradoxes do not hold in Direct by functional abstraction on Θ(Θ) 73 Logic. = f(Fix(f)) 70 Fix(f) ⇔ f(Fix(f)) Absolute Incompleteness Theorem by abstraction of equals ―that within us we always hear the call: here is the problem, search for the solution: you Liar Paradox can find it by pure thought, for in Oh what a tangled web we weave, mathematics there is no ignorabimus.” When first we practice to deceive! Hilbert [1900]74 Sir Walter Scott in ―Marmion‖
But paradoxes loom: the Liar Paradox goes back at least as far as the Greek philosopher Eubulides of Miletus who 73 For example, Russell‘s Paradox , Curry‘s Paradox, and the lived in the fourth century BC. It could be put as follows: Kleene-Rosser Paradox are not valid for theories in Direct Logic LiarProposition is defined to be the proposition “The because, in the empty theory ⊥ (that has no axioms beyond those negation of LiarProposition holds.” From its definition, LiarProposition holds if and only if it of Direct Logic): Russell’s Paradox: doesn‟t! Russell ≡ Fix(Diagonalize) The argument can be formalized using the fixed point theorem and the diagonal argument in the following way: where Diagonalize ≡ λ(s) ├⊥ s LiarProposition ≡ Fix(Diagonalize) ∴ Russell ├ Russell 71 ⇔ ⊥ where Diagonalize ≡ λ(s) s But presumably ├⊥ Russell is not Admissible for ⊥ Argument for the Liar Paradox72 LiarProposition ⇔ Fix(Diagonalize) Curry’s Paradox: Curry ≡ Fix(Diagonalize) ⇔ Diagonalize(Fix(Diagonalize)) where Diagonalize ≡ λ(s) s⇨ by the fixed point theorem ∴ Curry Curry ⇨ ⇔ λ(s) s (Fix(Diagonalize)) But presumably, in general Curry⇨ is not Admissible for ⊥ ⇔ Fix(Diagonalize)
⇔ LiarProposition Kleene-Rosser Paradox: KleeneRosser ≡ Diagonalize(Diagonalize) Therefore ⊢⊥ (LiarProposition ⇔ LiarProposition ) where Diagonalize ≡ λ(f) f(f) ∴ KleeneRosser ⇔ KleeneRosser
But presumably KleeneRosser is not Admissible for ⊥ 69 Where did the definition of Θ come from? First note that λ(x) (g(g))(x) = g(g) and consequently Paradox of Inferability Θ = λ(g) f(g(g)) Inferable ≡ Fix(Diagonalize) So Θ takes itself as an argument and returns the result of applying where Diagonalize ≡ λ(s) ├⊥ s f to the result of applying itself to itself! In this way a fixed point of f is constructed. ∴ Inferable ⇔ ├⊥Inferable
70 Note that equality (=) is not defined on abstract propositions But presumably├⊥Inferable is not Admissible for ⊥ (like Fix(f) ). Also note that logical equivalence () is not defined on concrete XML sentences (like Fix(f) ). 74 Reiterated in [Hilbert 1930] just before [Gödel 1931].
71 According to [Lewis 2004]: Note that Diagonalize always converges. Von Neumann often had insights into the repercussions of work that others would understand later; on hearing Gödel 72 As explained below, this argument is not valid in Direct Logic. present his results on formal incompleteness, he immediately forsook logic and said “it’s all over.” (emphasis added)
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Incompleteness of a theory T is defined to mean that there Theorem: Theories in Direct Logic are self-provably 75 is some proposition that is logically undecidable in T, i.e. absolutely incomplete. that the proposition cannot be proved and neither can its It is sufficient to self-prove that Uninferable is logically negation, i.e., a theory T is incomplete if and only if there T undecidable, i.e., is a logically undecidable proposition such that
1. ⊢ ⊬ Uninferable (⊬ ) (⊬ ) T T T T T The general heuristic for constructing such a sentence is 2. ⊢ ⊬ Uninferable T T T to construct a proposition that says the following: Proof of Theorem:76 This proposition is uninferable in T. 1) To prove: ⊬ Uninferable T T
⊢ (Uninferable ⇨ ⊬ Uninferable ) lemma Such a proposition (called Uninferable ) can be T T T T T constructed as follows using the fixed point theorem and Uninferable ⊢ ⊬ Uninferable T T T T diagonalization: 2-way deduction theorem
(⊢ Uninferable ) ⊢ (⊢ ⊬ Uninferable ) soundness Uninferable ≡ Fix(Diagonalize) T T T T T T T ⊢ ⊬ Uninferable T T T Argument for Self Infers Argument for Opposite where Diagonalize ≡ λ(s) ⊬ s T Diagonalize(s) is a sentence that says that 2) To prove: ⊬ Uninferable T T s is not inferable in T ⊢ (Uninferable ⇨ ├ Uninferable ) T T T T contrapositive of lemma Uninferable ├ (├ Uninferable ) The following lemma verifies that Uninferable has the T T T T T desired property: 2-way deduction theorem
(⊢ Uninferable ) ⊢ (⊢ ⊢ Uninferable ) Lemma: ⊢ (Uninferable ⇔ ⊬ Uninferable ) T T T T T T T T T T soundness Proof: (⊢ Uninferable ) ⊢ (⊢ Uninferable )) faithfulness T T T T T First show that ⊬ Uninferable is Admissible for T T T ⊢ ⊬ Uninferable Proof: We need to show the following: T T T Argument for Self Infers Argument for Opposite ((⊬ Uninferable )) ⊢ (⊢ (⊬ Uninferable )) T T T T T T which by double negation elimination is equivalent to However, as shown in the next section, a showing consequence of self-provable incompleteness is inconsistency. (⊢ Uninferable ) ⊢ (⊢ ⊢ Uninferable ) T T T T T T which follows immediately from adequacy.
Uninferable ⇔ Fix(Diagonalize) T ⇔ Diagonalize(Fix(Diagonalize))
logical fixed point theorem 75 Absolute incompleteness for a theory T is incompleteness that ⇔ λ(s) ⊬ s (Fix(Diagonalize)) T does not depend on the subject matter of T [Martin-Löf 1995, definition of Diagonalize Feferman 2006c].
⇔ ⊬ Fix(Diagonalize) 76 T Incompleteness of Principia Mathematica was proved informally using proof by contradiction in a stratified metatheory ⇔ ⊬ Uninferable T T in [Gödel 1931] with restrictive conditions. Then [Rosser 1936] informally proved incompleteness using proof by contradiction in ⇔ ⊬ Uninferable T T a stratified metatheory assuming consistency of Principia Mathematica. The formal proof below does not use stratified Admissibility of ⊬ Uninferable T T metatheories and uses self-refutation instead of proof by contradiction or.
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Absolute Inconsistency Theorem Logical necessity of inconsistency Have said—with pride in a mathematical All truth passes through three stages: discovery: „Look, this is how we produce a First, it is ridiculed. contradiction.‟ [Wittgenstein 1956] Second, it is violently opposed. Third, it is accepted as being self-evident. The test of a first-rate intelligence is the ability to Arthur Schopenhauer (1788-1860) hold two opposed ideas in the mind at the same It is worth noting that inconsistency about Uninferable time, and still retain the ability to function. T79 does not cause any particular problems for Direct Logic: --- F. Scott Fitzgerald in ―The Crack up‖ Because T is inconsistency tolerant, that T is Theorem: Theories in Direct Logic are self-provably inconsistent about Uninferable (by itself) should not 77 T inconsistent. affect other reasoning. Also the subject matter of
Uninferable is not of general interest in software It is sufficient to show that T proves both Uninferable T T engineering and should not affect reasoning about and its negation, i.e., current large software systems. So do software 1. ⊢ Uninferable engineers need to care that T is inconsistent about T T Uninferable as opposed to all the other 2. ⊢ Uninferable T T T inconsistencies of which they care about more?80 T Proof of theorem78 The logically necessary inconsistency concerning Uninferable is a nice illustration of how 1). ⊢ Uninferable is immediate from the T T T inconsistencies often arise in large software incompleteness theorem ⊢ ⊬ Uninferable and the systems: “there can be good arguments on both T T T sides for contradictory conclusions”. lemma ⊢ (Uninferable ⇔ ⊬ Uninferable ). T T T T A big advantage of inconstancy tolerance is that it makes fewer mistakes than classical logic when dealing 2) ⊢ Uninferable is immediate from ⊢ T T T with inconsistent theories. Since software engineers have Uninferable (above) and the contrapositive of the T to deal with theories chock full of inconsistencies, Direct
lemma ⊢ (Uninferable ⇔ ⊢ Uninferable ) Logic should be attractive. However, to make it relevant T T T T we need to provide them with tools that are cost effective. At first, TRUTH may seem like a desirable property for propositions in theories for large software systems. However, because a theory T is necessarily inconsistent about Uninferable , it is impossible to consistently assign T truth values to propositions of T. In particular it is 77 This theorem is closely related to dialetheism [Priest and impossible to consistently assign a truth value to the Routley 1989] which made the claim that mathematics is proposition Uninferable . If the proposition is assigned the T inconsistent (e.g. because of the Liar Paradox). Every theory of value TRUE, then (by the rules for truth values) it must Direct Logic is necessarily inconsistent because it self-infers the Gödelian paradoxical sentence, cf. [Routley 1979], [Priest and also be assigned FALSE and vice versa. Tanaka 2004], etc. 78 According to Wittgenstein, Let us suppose I prove the unprovability (in Russell‟s system
[Principia Mathematica] (PM)]) of P [⊢ ⊬ P] (where P PM PM is Uninferable the Gödel sentence of the system); then by PM this proof I have proved P [⊢ P]. Now if this proof were PM one in Russell‟s system—I should in this case have proved at once that it belonged [⊢ P] and did not belong [⊢ P] PM PM to Russell‟s system.—That is what comes of making up such 79 As pointed out above, the other standard paradoxes (Liar, sentences. But there is a contradiction here!—Well, then Russell, Curry, Kleene–Rosser, etc.) are blocked in Direct Logic. there is a contradiction here [in PM]. Does it do any harm here? [Wittgenstein, 1956, p. 51e] 80 Of course, there are other inconsistent propositions of the same ilk, cf., Rosser [1936].
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It is not obvious what (if anything) is wrong or how to Definition. fix it. Of course, this is contrary to the traditional view of Tarski who wrote: Argument Inferred propositions for T are I believe everybody agrees that one of the reasons those such that (├ ) ├ which may compel us to reject an empirical theory is T T the proof of its inconsistency. . . . It seems to me that the real reason of our attitude is...: We know (if only One might naively assume that all propositions are intuitively) that an inconsistent theory must contain Argument Inferred. The naive intuition is that if there is an false sentences. [Tarski 1944] argument for a proposition infers the proposition. However, On the other hand, Frege [1915] suggested that, in a as shown below, an argument inferred proposition must logically perfect language, the word ‗true‘ would not hold. Therefore every proposition could be inferred if all appear! According to McGee [2006], he argued that propositions are argument inferred!82 when we say that it is true that seawater is salty, we don‟t add anything to what we say when we say simply that seawater is salty, so the notion of truth, ArgInfers ≡ Fix(Diagonalize) in spite of being the central notion of [classical] where Diagonalize ≡ λ(s) (├ s ) ├ T T logic, is a singularly ineffectual notion. It is
surprising that we would have occasion to use such 83 an impotent notion, nevermind that we would regard Theorem : From that it hold in T that is Argument it as valuable and important. Inferred for T and (├ ArgInfers)├ is Admissible Why did Gödel and the logicians who followed him not T T for it follows that ├ go in this direction? Feferman [2006b] remarked on ―the T, T shadow of Hilbert that loomed over Gödel from the beginning to the end of his career.‖ Also Feferman [2006a] Proof: conjectured that ―Gödel simply found it galling all through Suppose that is argument inferred for T and his life that he never received the recognition from Hilbert (├ ArgInfers) ├ is Admissible for T. that he deserved.‖ Furthermore, Feferman maintained that T T It is sufficient to prove ├ ―the challenge remained well into his last decade for T Gödel to demonstrate decisively, if possible, why it is Lemma: ├ (ArgInfers⇔ ((├ ArgInfers)├ ))) necessary to go beyond Hilbert‟s finitism in order to T T T prosecute the constructive consistency program.‖ Indeed Proof: Gödel saw his task as being ―to find a consistency proof for ArgInfers ⇔ Fix(Diagonalize) arithmetic based on constructively evident though abstract ⇔ Diagonalize (Fix(Diagonalize)) principles‖ [Dawson 1997 pg. 263]. Gödel had a horror of ⇔λ(s) (├ s )├ (Fix(Diagonalize)) T T inconsistency because it was incompatible with his firmly ⇔ (├ Fix(Diagonalize) )├ held Platonism. T T ⇔ (├ ArgInfers)├ T T
Don’t believe every argument ⇔ ((├ ArgInfers) ├ ) T T 81 Don‟t believe everything you think . by Admissibility of (├ ArgInfers)├ T T Thomas Kida [2006]
In Direct Logic, simply because there is an argument for a proposition is not by itself sufficient to infer the proposition. Instead, arguments both for and against the proposition should be considered.
82 modulo questions of Admissibility 81 According to Frege‘s Grundgesetze , the laws of logic are ―the most general laws, which prescribe universally the way in which 83 Generalization of Löb‘s Theorem [Löb 1955]. Also see [Dean one ought to think if one is to think at all.‖ [Feferman 1999] and Kurokowa 2009].
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Proof of theorem84 Direct Logic makes the following contributions over Relevance Logic: Suppose (├ )├ T T • Boolean Equivalences We need to show that ├ T • Splitting (including Splitting by Negation) • Two-way Deduction Theorem (Natural Deduction) ├ (ArgInfers ├ ((├ ArgInfers) ├ ))) T T T T • Self-refutation lemma • Incompleteness Theorem self-provable ├ ((├ ArgInfers) ├ (├ ((├ ArgInfers) ├ )))) T T T T T T • Logical necessity of inconsistency soundness on above • Recursive decidability of Boolean fragment86
├ ((├ ArgInfers)├ (├ ((├ ├ ArgInfers)├ (├ )))) T T T T T T T T Concurrency is the Norm soundness on (├ ArgInfers)├ T T
├ ((├ ArgInfers)├ (├ ├ ArgInfers)) The distinction between past, present and T T T T T adequacy future is only a stubbornly persistent illusion. Albert Einstein ├ ((├ ArgInfers) ├ (├ )) detachment T T T T ├ ((├ ArgInfers) ├ ) Concurrency has now become the norm. However T T T transitivity on hypothesis nondeterminism came first.
├ ArgInfers transitivity on lemma T Nondeterministic computation ├ ├ ArgInfers adequacy on ├ ArgInfers T T T Several models of nondeterministic computation were ├ ((├ ├ ArgInfers) ├ (├ )) T T T T T developed including the following:
soundness on (├ ArgInfers) ├ Lambda calculus The lambda calculus can be viewed as T T the earliest message passing programming language ├ ((├ ArgInfers) ├ (├ )) T T T T [Hewitt, Bishop, and Steiger 1973]. For example the adequacy on ├ ArgInfers T lambda expression below implements a tree data structure when supplied with parameters for a leftSubTree and ├ ├ detachment on ├ ArgInfers T T T rightSubTree. When such a tree is given a parameter ├ faithfulness on ├ T T message "getLeft", it returns leftSubTree and likewise when given the message "getRight" it returns Contributions of Direct Logic rightSubTree. Direct Logic makes the following contributions over the λ leftSubTree, rightSubTree traditional Tarskian framework: λ message • Direct Inference (no contrapositive bug for inference) if message == “GetLeft") • Direct Argumentation (inference directly expressed) then leftSubTree • Inconsistency Tolerance else if message == “GetRight") Relevance Logic [Mares 2006] arose from attempts to then rightSubTree axiomatise the notion that an implication ⇨ should be However, the semantics of the lambda calculus were regarded to hold only if the hypothesis is ―relevant‖ to expressed using variable substitution in which the values of the conclusion . According to [Routley 1979], ―The parameters were substituted into the body of an invoked abandonment of disjunctive syllogism is indeed the lambda expression. The substitution model is unsuitable characteristic feature of the relevant logic solution to the for concurrency because it does not allow the capability of implicational paradoxes.‖85 Since Direct Logic sharing of changing resources. Inspired by the lambda calculus, the interpreter for the programming language incorporates disjunctive syllogism ((ΦΨ), ¬Φ ├ Ψ) and Lisp made use of a data structure called an environment so does not support disjunction introduction (Ψ ├ ΦΨ), it is that the values of parameters did not have to be substituted not a Relevance Logic. into the body of an invoked lambda expression. This allowed for sharing of the effects of updating shared data structures but did not provide for concurrency.
84 The proof is an adaptation for Direct Logic of [Löb 1955; Petri nets Prior to the development of the Actor model, Verbrugge 2003]. Petri nets were widely used to model nondeterministic
85 computation. However, they were widely acknowledged to Instead Relevance logics (e.g. R and RM) support disjunction introduction in the form ├(Ψ ⇨ ΦΨ) [Maddux 2007]. 86 However RM and S are recursively decidable [Deutsch 1985].
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have an important limitation: they modeled control flow The first (token) encountered (in a program) was looked but not data flow. Consequently they were not readily up in the dynamic context, to determine the receiver of composable thereby limiting their modularity. Hewitt the subsequent message. The name lookup began with pointed out another difficulty with Petri nets: simultaneous the class dictionary of the current activation. Failing action, i.e., the atomic step of computation in Petri nets is a there, it moved to the sender of that activation and so on transition in which tokens simultaneously disappear from up the sender chain. When a binding was finally found the input places of a transition and appear in the output for the token, its value became the receiver of a new places. The physical basis of using a primitive with this message, and the interpreter activated the code for that kind of simultaneity seemed questionable to him. Despite object's class.89 these apparent difficulties, Petri nets continue to be a popular approach to modeling nondeterminism, and are objects. The authors of Simula had considered making such still the subject of active research. primitives into objects but refrained largely for efficiency reasons. Java at first used the expedient of having both primitive Simula 67 pioneered using message passing for and object versions of integers, floating point numbers, etc. The computation, motivated by discrete event simulation C# programming language (and later versions of Java, starting applications. These applications had become large and with Java 1.5) adopted the more elegant solution of using boxing unmodular in previous simulation languages. At each time and unboxing, a variant of which had been used earlier in some Lisp implementations. step, a large central program would have to go through and update the state of each simulation object that changed 88 The Smalltalk system went on to become very influential, depending on the state of which ever simulation objects innovating in bitmap displays, personal computing, the class that it interacted with on that step. Kristen Nygaard and browser interface, and many other ways. Meanwhile the Actor Ole-Johan Dahl developed the idea (first described in an efforts at MIT remained focused on developing the science and IFIP workshop in 1967) of having methods on each object engineering of higher level concurrency that would update its own local state based on messages See the 2nd appendix of this paper on how Actors treated meta- from other objects. In addition they introduced a class circular evaluation differently than Smalltalk-72 and Briot [1988] structure for objects with inheritance. Their innovations for ideas that were developed later on how to incorporate some considerably improved the modularity of programs. Simula kinds of Actor concurrency into later versions of Smalltalk.
67 used nondeterministic coroutine control structure in its 89 simulations. According to the Smalltalk-72 Instruction Manual [Goldberg and Kay 1976]: There is not one global message to which all message "fetches" Smalltalk-72. Planner, Simula 67, Smalltalk-72 [Kay 1975; Ingalls 1983] and computer networks had previously (use of the Smalltalk symbols eyeball, ; colon, :, and open 표 used message passing. However, they were too colon, 표 ) refer; rather, messages form a hierarchy which we complicated to use as the foundation for a mathematical explain in the following way-- suppose I just received a theory of concurrency. Also they did not address message; I read part of it and decide I should send my friend a fundamental issues of concurrency. message; I wait until my friend reads his message (the one I sent him, not the one I received); when he finishes reading his Alan Kay was influenced by message passing in the message, I return to reading my message. I can choose to let my pattern-directed invocation of Planner in developing friend read the rest of my message, but then I cannot get the Smalltalk-71. Hewitt was intrigued by Smalltalk-71 but message back to read it myself (note, however, that this can be was put off by the complexity of communication that done using the Smalltalk object apply which will be discussed included invocations with many fields including global, later). I can also choose to include permission in my message to sender, receiver, reply-style, status, reply, operator my friend to ask me to fetch some information from my message selector, etc. and to give that in information to him (accomplished by 표 In November 1972 Kay visited MIT and discussed some including : or 표 in the message to the friend). However, of his ideas for Smalltalk-72 building on the Logo work of anything my friend fetches, I can no longer have. In other words, Seymour Papert and the "little person" metaphor of 1) An object (let's call it the CALLER) can send a message computation used for teaching children to program. to another object (the RECEIVER) by simply mentioning However, the message passing of Smalltalk-72 was quite the RECEIVER's name followed by the message. complex [Kay 1975]. Code in the language was viewed by 2) The action of message sending forms a stack of messages; the last message sent is put on the top. the interpreter as simply a stream of tokens.87 As Dan 88 3) Each attempt to receive information typically means Ingalls [1983] later described it: looking at the message on the top of the stack. 4) The RECEIVER uses the eyeball, , the colon, :, and the 87 표 Subsequent versions of the Smalltalk language largely open colon, 표 , to receive information from the message followed the path of using the virtual methods of Simula in the at the top of the stack. message passing structure of programs. However Smalltalk-72 5) When the RECEIVER completes his actions, the message made primitives such as integers, floating point numbers, etc. into at the top of the stack is removed and the ability to send
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Thus the message passing model in Smalltalk-72 was Arguments for incorporating unbounded closely tied to a particular machine model and nondeterminism programming language syntax that did not lend itself to Carl Hewitt [1985, 2006] argued against Dijkstra in concurrency.90 support of the Actor model: There is no bound that can be placed on how long Dijkstra argued against unbounded nondeterminism it takes a computational circuit called an arbiter to “Things are only impossible until they're not.” settle. Arbiters are used in computers to deal with Jean-Luc Picard in “Star Trek: The Next Generation” the circumstance that computer clocks operate asynchronously with input from outside, e.g., In theoretical Computer Science, unbounded keyboard input, disk access, network input, etc. So nondeterminism (sometimes called unbounded it could take an unbounded time for a message sent indeterminacy) is a property of concurrency by which the to a computer to be received and in the meantime amount of delay in servicing a request can become the computer could traverse an unbounded number unbounded as a result of arbitration of contention for of states. shared resources while still guaranteeing that the request Electronic mail enables unbounded will eventually be serviced. nondeterminism since mail can be stored on servers Edsger Dijkstra [1976] argued that it is impossible to indefinitely before being delivered. implement systems with unbounded nondeterminism Communication links to servers on the Internet can although the Actor model [Hewitt, Bishop, and Steiger be out of service indefinitely. 1973] explicitly supported unbounded nondeterminism. Nondeterministic automata Nondeterministic Turing machines have only bounded nondeterminism. Sequential programs containing guarded and receive messages returns to the CALLER. The commands as the only sources of nondeterminism have RECEIVER may return a value to be used by the only bounded nondeterminism [Dijkstra 1976] because CALLER. 6) This sequence of sending and receiving messages, viewed choice nondeterminism is bounded. Gordon Plotkin [1976] here as a process of stacking messages, means that each gave a proof as follows: message on the stack has a CALLER (message sender) Now the set of initial segments of execution sequences and RECEIVER (message receiver). Each time the of a given nondeterministic program P, starting from a RECEIVER is finished, his message is removed from the given state, will form a tree. The branching points will stack and the CALLER becomes the current RECEIVER. correspond to the choice points in the program. Since The now current RECEIVER can continue reading any there are always only finitely many alternatives at each information remaining in his message. choice point, the branching factor of the tree is always 7) Initially, the RECEIVER is the first object in the message finite. That is, the tree is finitary. Now König's lemma typed by the programmer, who is the CALLER. says that if every branch of a finitary tree is finite, then 8) If the RECEIVER's message contains a eyeball, ; colon, so is the tree itself. In the present case this means that 표 if every execution sequence of P terminates, then there :, or open colon, 표 , he can obtain further information are only finitely many execution sequences. So if an from the CALLER's message. Any information output set of P is infinite, it must contain a successfully obtained by the RECEIVER is no longer nonterminating computation. available to the CALLER. 9) By calling on the object apply, the CALLER. can give the RECEIVER the right to see all of the CALLER's Indeterminacy in concurrent computation versus remaining message. The CALLER can no longer get nondeterministic automata information that is read by the RECEIVER; he can, Will Clinger [1981] provided the following analysis of the however, read anything that remains after the RECEIVER above proof by Plotkin: completes its actions. 10) There are two further special Smalltalk symbols useful in This proof depends upon the premise that if every node sending and receiving messages. One is the keyhole, , x of a certain infinite branch can be reached by some that lets the RECEIVER ―peek‖ at the message. It is the computation c, then there exists a computation c that 표 goes through every node x on the branch. ... Clearly same as the 표 except it does not remove the information from the message. The second symbol is the hash mark, #, this premise follows not from logic but rather from the placed in the message in order to send a reference to the interpretation given to choice points. This premise fails next token rather than the token itself. for arrival nondeterminism [in the arrival of messages in the Actor model] because of finite delay [in the 90 Although Smalltalk-72 was bootstrapped on itself, the language arrival of messages]. Though each node on an infinite constructs were not formally defined as objects that respond to branch must lie on a branch with a limit, the infinite Eval messages (see [Hewitt 2009d]). branch need not itself have a limit. Thus the existence
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of an infinite branch does not necessarily imply a other inference capabilities, Direct Logic is a very nonterminating computation. powerful Logic Programming language.
Bounded nondeterminism in CSP Unbounded nondeterminism in ActorScript Consider the following program written in CSP [Hoare Nevertheless, there are concurrent programs that are not 1978]: equivalent to any Direct Logic program. For example in [X:: Z!stop( ) the Actor model, the following concurrent program in || ActorScriptTM [Hewitt 2009c] will return an integer of Y :: guard: boolean; guard := true; unbounded size is not equivalent to any Direct Logic *[guard Z!go( ); Z?guard] expression (for reasoning see below) || Unbounded Z :: n: integer; n:= 0; {behavior continue: boolean; continue := true; [self start] when a start message is received *[X?stop( ) continue := false; 퐜 [] {let ( = let c be a Counter 퐂퐨퐮퐧퐭퐞퐫 Y?go( ) n := n+1; Y!continue create SimpleCounter(n=0,continue=true)) ] which is a new SimpleCounter with ] count equal 0 and continue equal true According to Clinger [1981]: This program illustrates global nondeterminism, since the {[c stop], send a stop message to c and nondeterminism arises from incomplete specification of the [c go]}}} concurrently return the value of timing of signals between the three processes X, Y, and Z. The sending a go message to c repetitive guarded command in the definition of Z has two alternatives: either the stop message is accepted from X, in SimpleCounter which case continue is set to false, or a go message is {serializer 퐧 accepted from Y, in which case n is incremented and Y is sent n is the current count 퐈퐧퐭퐞퐠퐞퐫 the value of continue. If Z ever accepts the stop message from X, then X terminates. Accepting the stop causes continue to be 퐜퐨퐧퐭퐢퐧퐮퐞 continue is true until set to false, so after Y sends its next go message, Y will receive 퐈퐧퐭퐞퐠퐞퐫 false as the value of its guard and will terminate. When both X stop has been received and Y have terminated, Z terminates because it no longer has ------live processes providing input. implements Counter As the author of CSP points out, therefore, if the repetitive implements the Counter93 interface guarded command in the definition of Z were required to be [self go] when an go message is received fair, this program would have unbounded nondeterminism: it {match continue would be guaranteed to halt but there would be no bound on true : continue is true the final value of n91. In actual fact, the repetitive guarded commands of CSP are not required to be fair, and so the {future [self go] alsoBecome (n=n+1)} program may not halt [Hoare 1978]. This fact may be return a future of sending a confirmed by a tedious calculation using the semantics of CSP go message to this counter (self) also [Francez, Hoare, Lehmann, and de Roever 1979] or simply by update n by incrementing by 1 noting that the semantics of CSP is based upon a conventional false : continue is false power domain and thus does not give rise to unbounded {n}} return the count n 92 nondeterminism. [self stop] when a stop message is received Since it includes the nondeterministic λ calculus, direct {void alsoBecome (continue=false)}} inference, and mathematical induction in addition to its return void also update continue to be false
91 Of course, n would not survive the termination of Z and so the value cannot actually be exhibited after termination!
92 Dijkstra had convinced Hoare that unbounded nondeterminism 93 The Counter interface can be defined as follows: is impossible to implement and consequently the semantics of Counter Counter is defined to be CSP had bounded nondeterminism [Francez, Hoare, Lehmann, {interface an interface and de Roever 1979]. [퐬퐞퐥퐟 퐠퐨 ] a go message returns void This error was corrected in the revised version of CSP ([Hoare 푣표푖푑 1985; Roscoe 2005]) to explicitly provide unbounded [퐬퐞퐥퐟 퐬퐭퐨퐩 ] } a stop message returns an Integer nondeterminism. 퐈퐧퐭퐞퐠퐞퐫
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By the semantics of the Actor model of computation In this way we have proved that the Procedural [Clinger 1981] [Hewitt 2006], the result of evaluating the Embedding of Knowledge paradigm is strictly more expression [Unbounded start]94 is an integer of general than the Logic Programming paradigm. unbounded size. Computation is not subsumed by logical deduction Bounded Nondeterminism of Direct Logic The notion of computation has been evolving for a long But there is no Direct Logic expression that is equivalent to time. One of the earliest examples was Euclid‘s GCD [Unbounded start] for the following reason: algorithm. Next came mechanical calculators of various An expression will be said to always converge (written kinds. These notions were formalized in the Turing as ) if and only if every reduction path terminates. I.e., Machines, the lambda calculus, etc. paradigm that focused there is no function f(Expressions) such that on the ―state‖ of a computation that could be logically f(0)= and (n: f(n) f(n+1) ) inferred from the ―previous‖ state. Scott and Strachey [1971] furthered the research program of attempting to where the symbol is used for reduction in the reduce all computation to the lambda calculus culminating nondeterministic λ calculus (see the Appendix). For in Milne and Strachey [1976].97 example (λ(x) 0 | x(x)) (λ(x) 0 | x(x))95 because there is a nonterminating path. Computational objects in process calculi are anonymous, Theorem: Bounded Nondeterminism of Direct Logic. If and communicate by sending messages either through an expression in Direct Logic always converges, then named channels (synchronous or asynchronous), or via there is a bound Bound on the number to which it can ambients (which can also be used to model channel-like converge. I.e., communications [Cardelli and Gordon 1998]). In contrast, n: (n ⇨ n≤Bound) Actors in the Actor model possess an identity, and communicate by sending messages to the mailing Consequently there is no Direct Logic program equivalent addresses of other Actors (this style of communication can to [Unbounded start] because it has unbounded also be used to model channel-like communications). nondeterminism whereas every Direct Logic program has For example, communication in the π –calculus [Milner 1993] bounded nondeterminism.96 takes the following form: input: u(x).P is a process that gets a message from a 94 which sends Unbounded a start message communication channel named u before proceeding as P, binding the message received to the identifier x. In ActorScript, this can be modeled as follows: 95 Note that there are two bodies (separated by “|”) in each of the let x=[u get]; P λ expressions which provides for nondeterminism. output: ū
96 communication channel u before proceeding as P. In The status of unbounded nondeterminism varies in process ActorScript, this can be modeled as follows: calculi (e.g. [Milner 1993; Cardelli and Gordon 1998]). There are {[u put: m]; P} many similarities between the Actor Model and process calculi, The rest of the π -calculus can be modeled using a two-phase but also several differences (some philosophical, some technical): commit protocol [Knabe 1992; Reppy, Russo, and Xiao 2009]. There is only one Actor model (although it has numerous
formal systems for design, analysis, verification, 97 In the 1960‘s at the MIT AI Lab, a culture developed around modeling, etc.); there are numerous process calculi, ―hacking‖ that concentrated on remarkable feats of programming developed for reasoning about a variety of different kinds [Levy 1984]. Growing out of this tradition, Gerry Sussman and of concurrent systems at various levels of detail (including Guy Steele decided to try to understand Actors by reducing them calculi that incorporate time, stochastic transitions, or to machine code that they could understand and so developed a constructs specific to application areas such as security ―Lisp-like language, Scheme, based on the lambda calculus, but analysis). extended for side effects, multiprocessing, and process synchronization.‖ [Sussman and Steele 2005] Unfortunately, their The Actor model was inspired by the laws of physics and reductionist approach included primitives like depends on them for its fundamental axioms, i.e. physical START!PROCESS, STOP!PROCESS and laws (see Actor model theory); the process calculi were EVALUATE!UNINTERRUPTIBLEY that had the following originally inspired by algebra [Milner 1993]. explanation: Semantics of the Actor model is based on message event This is the synchronization primitive. It evaluates an orderings in the Computational Representation Theorem. expression uninterruptibly; i.e. no other process may run until the expression has returned a value. Semantics of process calculi are based on structural Of course, the above reductionist approach is very unsatisfactory congruence in various kinds of bisimulations and because it missed a crucial aspect of the Actor model: the arrival equivalences. ordering of messages.
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The invention of digital computers caused a decisive Robert Kowalski developed the thesis that ―computation paradigm shift when the notion of an interrupt was could be subsumed by deduction” [Kowalski 1988a] that invented so that input that arrived asynchronously from he states was first proposed by Hayes [1973] in the form outside could be incorporated in an ongoing computation. ―Computation = controlled deduction.”99 [Kowalski 1979] The break was decisive because asynchronous Kowalski forcefully stated: communication cannot be implemented by Turing There is only one language suitable for machines etc. because the order of arrival of messages representing information -- whether declarative or cannot be logically inferred.98 procedural -- and that is first-order predicate Actors [Hewitt, Bishop, and Steiger 1973] was a new logic. There is only one intelligent way to process model of computation based on message passing in which information -- and that is by applying deductive there is no global state and unbounded nondeterminism is inference methods. [Kowalski 1980] modeled. Furthermore, unbounded nondeterminism is a The gauntlet was officially thrown in The Challenge of fundamental property of the Actor Model because it Open Systems [Hewitt 1985] to which [Kowalski 1988b] provides a guarantee of service for shared resources. In replied in Logic-Based Open Systems. This was followed previous models of computation with bounded up with [Hewitt and Agha 1988] in the context of the nondeterminism, it was possible for a request to a shared Japanese Fifth Generation Project. resource to never receive service because it was possible The Hayes-Kowalski thesis was valuable in that it that a nondeterministic choice would always be made to motivated further research to characterize exactly which service another request instead. Message passing has computations could be performed by Logic Programming. become the foundation of many-core and client-cloud However, contrary to the quotations (above) by Kowalski computing. and Hayes, computation in general cannot be subsumed by deduction and contrary to the quotation (above) attributed to Hayes, computation in general is not controlled In summary, Sussman and Steele [1975] mistakenly concluded deduction. In fact, Logic Programming is not ―we discovered that the 'Actors' and the lambda expressions were computationally universal as explained below. identical in implementation.‖ The actual situation is that the lambda calculus is capable of expressing some kinds of sequential Arrival order indeterminacy and parallel control structures but, in general, not the concurrency Hewitt and Agha [1991] and other published work argued expressed in the Actor model. On the other hand, the Actor model is capable of expressing everything in the lambda calculus and that mathematical models of concurrency did not determine particular concurrent computations as follows: more. 100 The Actor Model makes use of arbitration for 98 Computation was conceived in terms of nondeterministic determining which message is next in the arrival order of computation (e.g. Turing machines, Post productions, the lambda an Actor that is sent multiple messages concurrently. For calculus, Petri nets, nondeterministic simulations, etc.) in which example Arbiters can be used in the implementation of the each computational step changed the global state. However, it arrival order of messages sent to an Actor which are was well known that nondeterministic state machines have subject to indeterminacy in their arrival order. Since arrival bounded nondeterminism, i.e., if a machine is guaranteed to halt orders are in general indeterminate, they cannot be then it halts in a bounded number of states. Unbounded deduced from prior information by mathematical logic nondeterminism may at first seem like a rather esoteric property alone. Therefore mathematical logic cannot implement that is of no practical interest. However, this turns out not to be concurrent computation in open systems. the case because the ability to guarantee that a shared server will In concrete terms for Actor systems, typically we cannot respond to the requests that it receives requires unbounded observe the details by which the arrival order of messages nondeterminism. for an Actor is determined. Attempting to do so affects the However, there is no bound that can be placed on how long it results and can even push the indeterminacy elsewhere. takes a computational circuit called an arbiter to settle. Arbiters are used in computers to deal with the circumstance that computer Instead of observing the internals of arbitration processes clocks operate asynchronously with input from outside, e.g. of Actor computations, we await outcomes. Indeterminacy keyboard input, disk access, network input, etc. So it could take in arbiters produces indeterminacy in Actors. The reason an unbounded time for a message sent to a computer to be that we await outcomes is that we have no alternative received and in the meantime the computer could traverse an because of indeterminacy. unbounded number of states.98 Thus computers have the property It is important to be clear about the basis for the of unbounded nondeterminism. So there is an inconsistency published claim about the limitation of mathematical logic. between the nondeterministic state model of computation and the circuit model of arbiters. 99 Of course the same limitation applies to the Abstract State This thesis was also implicit in one interpretation of Cordell Machine (ASM) model [Blass, Gurevich, Rosenzweig, and Green‘s earlier work [Green 1969]. Rossman 2007a, 2007b; Glausch and Reisig 2006]. In the presence of arbiters, the global states in ASM are mythical. 100 Actors are the universal primitives of concurrent computation.
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It was not that individual Actors could not in general be One solution is to develop a concurrent variant of the implemented in mathematical logic. The claim is that Lisp meta-circular definition [McCarthy, Abrahams, because of the indeterminacy of the physical basis of Edwards, Hart, and Levin 1962] that was inspired by communication in the Actor model, no kind of inferential Turing's Universal Machine [Turing 1936]. If exp is a Lisp mathematical logic can deduce the order or arrival of future expression and env is an environment that assigns values 101 messages and the resulting computational steps. to identifiers, then the procedure EVAL with arguments exp and env evaluates exp using env. In the concurrent Computational Representation Theorem variant, Eval(env) is a message that can be sent to exp to What does the mathematical theory of Actors have to say cause exp to be evaluated. Using such messages, modular about this? A closed system is defined to be one which meta-circular definitions can be concisely expressed in the does not communicate with the outside. Actor model Actor model for universal concurrent programming theory provides the means to characterize all the possible languages [Hewitt 2009c]. computations of a closed system in terms of the Computational Representation Theorem [Clinger 1982; Scientific Community Metaphor Hewitt 2006]: The important thing in science is not so much to obtain new facts as to discover The denotation DenoteS of a closed system S represents new ways of thinking about them. all the possible behaviors of S as i Sir William Bragg DenoteS = ⊔iω ProgressionS (⊥S) Building on the Actor model of concurrent computation, where ProgressionS is an approximation function that Kornfeld and Hewitt [1981] developed fundamental takes a set of partial behaviors to their next stage and principles for Logic Programming in the Scientific ⊥S is the initial behavior of S. Community Metaphor [Hewitt 2006 2008b]: Monotonicity: Once something is published it In this way, the behavior of S can be mathematically cannot be undone. Scientists publish their results so characterized in terms of all its possible behaviors they are available to all. Published work is collected (including those involving unbounded nondeterminism). and indexed in libraries. Scientists who change their mind can publish later articles contradicting earlier Although DenoteS is not an implementation of S, it can be ones. However, they are not allowed to go into the used to prove a generalization of the Church-Turing- libraries and “erase” old publications. Rosser-Kleene thesis [Kleene 1943]: Concurrency: Scientists can work concurrently, Enumeration Theorem: If the primitive Actors of a overlapping in time and interacting with each other. closed Actor System S are effective, then the possible Commutativity: Publications can be read regardless outputs of S are recursively enumerable. of whether they initiate new research or become Proof: Follows immediately from the relevant to ongoing research. Scientists who become Representation Theorem. interested in a scientific question typically make an effort to find out if the answer has already been The upshot is that concurrent systems can be published. In addition they attempt to keep abreast represented and characterized by logical deduction of further developments as they continue their work. but cannot be implemented. Thus, the following Sponsorship: Sponsors provide resources for practical problem arose: computation, i.e., processing, storage, and communications. Publication and subscription How can practical programming languages be require sponsorship although sometimes costs can rigorously defined since the proposal by Scott and be offset by advertising. Strachey [1971] to define them in terms lambda Pluralism: Publications include heterogeneous, calculus failed because the lambda calculus cannot overlapping and possibly conflicting information. implement concurrency? There is no central arbiter of truth in scientific communities. Skepticism: Great effort is expended to test and validate current information and replace it with better information. 101 [Goldin and Wegner 2008] may seem superficially similar. Provenance: The provenance of information is Unfortunately, it failed to comprehend the Actor model or many carefully tracked and recorded. of the previously published arguments (e.g. [Hewitt, Bishop and Steiger 1973], [Hewitt 1977], and [Hewitt and Agha 1988]) Rapid recovery: Instead of making immense effort summarized in this paper. to maintain absolute consistency and hide all
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failures, resources are deployed to recover as rapidly Undecidability. Outcomes may be recursively as possible. undecidable. Initial experiments implementing the Scientific Inconsistency tolerance. There are typically good Community Metaphor revolved around the development of arguments for both sides of contradictory a programming language named Ether that had procedural conclusions. plans to process goals and assertions concurrently and Necessary Inconsistency. Theories of Direct Logic dynamically created new plans during program execution are necessarily inconsistent. [Kornfeld and Hewitt 1981]. Ether also addressed issues of Concurrency. Other concurrently operating conflict and contradiction with multiple sources of system components may block, interfere with, or knowledge and multiple viewpoints. revert possible outcomes. Ether used viewpoints to relativise information in publications. However a great deal of information is shared Indeterminacy. Because of concurrency, the outcomes may be physically indeterminate. across viewpoints. So Ether made use of inheritance so that information in a viewpoint could be readily used in other Entanglement. The very process of reasoning viewpoints. Sometimes this inheritance is not exact as about possible outcomes can affect the outcomes. when the laws of physics in Newtonian mechanics are Partiality. There might not be sufficient derived from those of Special Relativity. In such cases, information or resources available to infer Ether used translation instead of inheritance building on outcomes. work by Imre Lakatos [1976] who studied very Nonuniversality. Logic Programs are not sophisticated kinds of translations of mathematical computationally universal because they cannot theorems (e.g., the Euler formula for polyhedra). Later implement some concurrent programs. Bruno Latour [1988] analyzed translation in scientific These limitations lead to an admission of logical communities. powerlessness: Viewpoints were used to implement natural deduction In general, a component of a large software system is (Fitch [1952]) in Ether. In order to prove a goal of the form logically powerless over the outcome of its actions. ├V (P ⇨ Q) for a viewpoint V, it is sufficient to create a This admission of powerlessness needs to become part of 103 new viewpoint V' that inherits from V, assert ├V’ P, and the common sense of large software systems. then prove ├V’ Q. Hierarchical viewpoints of this kind were introduced into Planner-like languages in the context Work to be done mechanism of QA-4 [Rulifson, Derksen, and Waldinger 1973]. The best way to predict the future is to invent it. Resolving issues among viewpoints requires negotiation Alan Kay as studied in the sociology and philosophy of science. There is much theoretical work to be done to further develop Direct Logic. The admission of logical powerlessness The nontriviality104 of Direct Logic needs to be proved relative to the consistency of classical We must know, we will know. mathematics. Hilbert (1930) (also on his tombstone) In this regard Direct Logic is consonant with
Bourbaki: Descartes [1644] put forward the thesis that self-reference Absence of contradiction, in mathematics as conveys power, specifically the power of existence, as in “I 102 a whole or in any given branch of it, … think, therefore I am.” Direct inference conveys ability appears as an empirical fact, rather than as a for large software systems to directly reason about the metaphysical principle. The more a given possible outcomes of their actions. However there are branch has been developed, the less likely it limitations to logical inference including the following: becomes that contradictions may be met with Admissibility. It may not be safe to use logical in its farther development.105 roundtripping on propositions (about outcomes) that are not admissible. 103 Incompleteness. It may be impossible to logically Admission of powerlessness is the beginning of Step 1 in 12- prove or disprove outcomes. step programs of recovery from addiction, first developed by Alcoholics Anonymous, e.g., see Wilson [1952].
104 Nontriviality means that not everything can be proved. 102 From the Latin, ―Cogito ergo sum.” 105 [André Weil 1949] speaking as a representative of Bourbaki
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Thus the long historical failure to find an works with the blood of his heart. explosion in the methods used by Direct Logic Boltzmann can be considered to be strong evidence of its nontriviality. A number of Logicists feel threatened by the results in this Inconsistency tolerance of theories of Direct paper. Logic needs to be formally defined and proved. Some would like to stick with just classical logic Church remarked as follows concerning a and not consider inconsistency tolerance.108 Foundation of Logic that he was developing: Our present project is to develop the consequences Some would like to stick with the Tarskian of the foregoing set of postulates until a stratified theories and not consider direct contradiction is obtained from them, or until the inference. development has been carried so far consistently as Some would like to stick with just Logic to make it empirically probable that no Programming (e.g. nondeterministic Turing contradiction can be obtained from them. And in Machines, λ calculus, etc.) and not consider this connection it is to be remembered that just such concurrency. empirical evidence, although admittedly inconclusive, is the only existing evidence of the And some would like to have nothing to do with any of the freedom from contradiction of any system of above! However, the results in this paper (and the driving mathematical logic which has a claim to adequacy. technological and economic forces behind them) tend to [Church 1933]106 push towards inconsistency tolerance, direct inference, and Direct Logic is in a similar position except that the concurrency. [Hewitt 2008a] task is to demonstrate inconsistency tolerance Software engineers for large software systems often have instead of consistency. Also Direct Logic has good arguments for some proposition P and also good overcome many of the problems of Church‘s arguments for the negation of P, which is troubling. So Foundation of Logic. what do large software manufacturers do? If the problem is serious, they bring it before a committee of stakeholders to Inconsistencies such as the one about Uninferable T try and sort it out. In many particularly difficult cases the are relatively benign in the sense that they lack resulting decision has been to simply live with the problem significant consequences to software engineering. for a while. Consequently, large software systems are
Other propositions (such as ├ 1=0) are more shipped to customers with thousands of known T malignant because they can be used to infer that all inconsistencies of varying severity. The challenge is to try integers are equal to 0 using induction. To address malignant propositions, deeper investigations of 108 In 1994, Alan Robinson noted that he has ―always been argumentation using ╟ 107 must be undertaken in T a little quick to make adverse judgments about what I like which the provenance of information will play a to call ‗wacko logics‘ especially in Australia…I conduct central role. See [Hewitt 2009b]. my affairs as though I believe … that there is only one fundamental principles of argumentation or many- logic. All the rest is variation in what you‘re reasoning core semantic engines? See [Hewitt 2008a, 2008b]. about, not in how you‘re reasoning … [Logic] is Tooling for Direct Logic needs to be developed to immutable.‖ (quoted in Mackenzie [2001] page 286) support large software systems. See [Hewitt 2009b]. On the other hand Richard Routley noted: … classical logic bears a large measure of responsibility for the growing separation between philosophy and logic which Conclusion there is today… If classical logic is a modern tool What the poet laments holds for the inadequate for its job, modern philosophers have shown a mathematician. That he writes his classically stoic resignation in the face of this inadequacy. They have behaved like people who, faced with a device, designed to lift stream water, but which is so badly designed that it spills most of its freight, do not set themselves to the design of a better model, but rather devote much of their energy to constructing ingenious arguments to convince 106 The difference between the time that Church wrote the above themselves that the device is admirable, that they do not and today is that the standards for adequacy have gone up need or want the device to deliver more water; that there is dramatically. Direct Logic must be adequate to the needs of nothing wrong with wasting water and that it may even be reasoning about large software systems. Roundtripping is one of desirable; and that in order to “improve” the device they the biggest challenges to proving that Direct Logic is would have to change some features of the design, a thing inconsistency tolerant. which goes totally against their engineering intuitions and which they could not possibly consider doing. [Routley 107 ╟ means that is an argument for in T 2003] T
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to keep the situation from getting worse as systems of inference has been through Eubulides [4th century BC], continue to increase in complexity. Cantor [1890], Zermelo [1908], Russell [1908], Gödel Direct Logic has important advantages over previous [1931], Rosser [1936], Turing [1936], Curry [1942], Löb proposals (e.g. Relevance Logic) to more directly connect [1955], etc. leading ultimately to logically necessary inconsistency. antecedents to consequences in reasoning. These The concept of TRUTH has already been hard hit by the advantages include: pervasive inconsistencies of large software systems. using natural deduction reasoning Accepting necessary logical inconsistency would be preserving the standard Boolean equivalences another nail in its coffin. Ludwig Wittgenstein (ca. 1939) (double negation, De Morgan, etc.) said ―No one has ever yet got into trouble from a being able to more safely reason about the contradiction in logic.‖ to which Alan Turing responded mutually inconsistent data, code, specifications, ―The real harm will not come in unless there is an application, in which case a bridge may fall down.‖[Holt and test cases of client cloud computing 2006] It seems that we may now have arrived at the absence of contrapositive inference bug remarkable circumstance that we can‘t keep our systems having an intuitive deduction theorem that from crashing without allowing contradictions into our connects logical implication with inference. logic! inference in Boolean109 Direct Logic is This paper also proves that Logic Programming is not 110 computationally universal in that there are concurrent recursively decidable programs for which there is no equivalent in Direct Logic. Direct Logic preserves as much of classical logic as Thus the Logic Programming paradigm is strictly less possible given that it is based on direct inference. general than the Procedural Embedding of Knowledge Logicists are now challenged as to whether they agree paradigm. that Of course the results of this paper do not diminish the Inconsistency is the norm. importance of logic.111 There is much work to be done!112 Direct inference is the norm. Our everyday life is becoming increasingly dependent on large software systems. And these systems are becoming Logic Programming is not computationally increasingly permeated with inconsistency and universal. concurrency. As these pervasively inconsistent concurrent systems become a major part of the environment in which we A big advantage of inconsistency tolerant logic is that it live, it becomes an issue of common sense how to use makes fewer mistakes than classical logic when dealing with inconsistent theories. Since software engineers have to deal with theories chock full of inconsistencies, Direct 111 In a similar way, the incompleteness theorems did not Logic should be attractive. However, to make it relevant diminish the importance of logic although they also caused we need to provide them with tools that are cost effective. concern among some Logicists. For example Paul Bernays This paper develops a very powerful formalism (called (David Hilbert‘s assistant) wrote ―I was doubtful already Direct Logic) that incorporates the mathematics of sometime before [1931] about the completeness of the formal Computer Science and allows direct inference for almost system [for number theory], and I uttered [my doubts] to Hilbert, all of classical logic to be used in a way that is suitable for who was much angry … Likewise he was angry at Gödel‟s Software Engineering. results.‖ (quoted in Dawson [1998]) Gödel and Rosser proved that it is not possible to decide In fact, Hilbert never became reconciled with incompleteness all mathematical questions by inference. However, the incompleteness theorem relies on the assumption of as evidenced by the last two paragraphs of Hilbert's preface to consistency! This paper proves a generalization of the [Hilbert and Bernays 1934] (translation by Wilfried Sieg): Gödel/Rosser incompleteness theorem: a theory in Direct ―This situation of the results that have been achieved thus Logic is incomplete without relying on the assumption of far in proof theory at the same time points the direction for consistency. However, there is a further consequence. the further research with the end goal to establish as Although the classical mathematical fragment of Direct consistent all our usual methods of mathematics. Logic is evidently consistent, since the Gödelian With respect to this goal, I would like to emphasize the paradoxical proposition is self-provable, every theory in following: the view, which temporarily arose and which Direct Logic is inconsistent! The mathematical exploration maintained that certain recent results of Gödel show that my proof theory can't be carried out, has been shown to be erroneous. In fact that result shows only that one must 109 Boolean propositions use only the connectives for exploit the finitary standpoint in a sharper way for the conjunction, disjunction, implication, and negation. farther reaching consistency proofs.”
110 In this way Direct Logic differs from Relevance Logic 112 In the film Dangerous Knowledge [Malone 2006], explores because a wide range of Boolean Relevance Logics (except for the history of previous crises in the foundations for the logic of RM and S) are recursively undecidable. [Deutsch 1985] knowledge focusing on the ultimately tragic personal outcomes for Cantor, Boltzmann, Gödel, and Turing.
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them effectively. We will need sophisticated software dinner with Harvey Friedman at Chez Panisse after his 2nd systems to help people understand and apply the Tarski lecture. principles and practices suggested in this paper. Creating Jeremy Forth, Tim Hinrichs, Fanya S. Montalvo, and this software is not a trivial undertaking! Richard Waldinger provided helpful comments and suggestions on the logically necessary inconsistencies in theories of Direct Logic. Rineke Verbrugge provided Acknowledgements valuable comments and suggestions at MALLOW‘07. Sol Feferman, Mike Genesereth, David Israel, Bill Jarrold, Mike Genesereth and Gordon Plotkin kindly hosted my Ben Kuipers, Pat Langley, Vladimir Lifschitz, Frank lectures at Stanford and Edinburgh, respectively, on ―The McCabe, John McCarthy, Fanya S. Montalvo, Peter Logical Necessity of Inconsistency‖. Inclusion of Cantor‘s Neumann, Ray Perrault, Natarajan Shankar, Mark Stickel, diagonal argument as motivation as well as significant Richard Waldinger, and others provided valuable feedback improvements in the presentation of the incompleteness at seminars at Stanford, SRI, and UT Austin to an earlier and inconsistency theorems were suggested by Jeremy version of the material in this paper. For the AAAI Spring Forth. John McCarthy pointed to the distinction between Symposium‟06, Ed Feigenbaum, Mehmet Göker, David Logic Programming and the Logicist Programme for Lavery, Doug Lenat, Dan Shapiro, and others provided Artificial Intelligence. Reviewers at JAIR made useful valuable feedback. At MIT Henry Lieberman, Ted Selker, suggestions. Mark S. Miller made important suggestions Gerry Sussman and the members of Common Sense for improving the meta-circular definition of ActorScript. Research Group made valuable comments. Reviewers for Comments by Michael Beeson helped make the AAMAS ‟06 and „07, KR‟06, COIN@AAMAS‟06 and presentation of Direct Logic more rigorous. Conversations IJCAR‟06 made suggestions for improvement. with Jim Larson helped clarify the relationship between In the logic community, Mike Dunn, Sol Feferman, classical logic and the inconsistency tolerant logic. An Mike Genesereth, Tim Hinrichs, Mike Kassoff, John anonymous referee of the Journal of Logic and McCarthy, Chris Mortensen, Graham Priest, Dana Scott, Computation made a useful comment. John-Jules Meyer Richard Weyhrauch and Ed Zalta provided valuable and Albert Visser provided helpful advice and suggestions. feedback Comments by Mike Genesereth, Eric Kao, and Mary-Anne Dana Scott made helpful suggestions on logical Williams at my Stanford Logic Group seminar ―Inference roundtripping and incompleteness. Richard Waldinger in Boolean Direct Logic is Recursively Decidable‖ on 18 provided extensive suggestions that resulted in better November 2009 greatly improved the explanation of direct focusing a previous version of this paper and increasing its inference. Discussions at my seminar ―Direct Inference for TM readability. Sol Feferman reminded me of the connection Direct Logic ‖ at SRI hosted by Richard Waldinger on 7 between Admissibility and 1. Discussion with Pat Hayes January 2010 helped improve the presentation of Direct and Bob Kowalski provided insight into the early history Logic. of Prolog. Communications from John McCarthy and Marvin Minsky suggested making common sense a focus. Mike Dunn collaborated on looking at the relationship of Bibliography the Boolean Fragment of Direct Logic to R-Mingle. Greg Hal Abelson and Gerry Sussman Structure and Restall pointed out that Direct Logic does not satisfy some Interpretation of Computer Programs 1984. Relevantist principles. 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Appendix. Additional Principles of Direct Logic (if False then E1 else E2) E2 Finally there are simple ideas of which no (1 2) (2 3)) ⇨ (1 3) definition can be given; there are also axioms (λ(x) F(x))) F() reduction
or postulates, or in a word primary 1 | 2 1 principles, which cannot be proved and have nondeterministic reduction to first alternative no need of proof. Gottfried Leibniz | 1 2 2 This appendix contains additional principles of Direct nondeterministic reduction to second alternative Logic. F1 F2 ⇨ F1() F2() an application reduces if its operator reduces
Equality 1 2 ⇨ F(1) F(2) Note that, in Direct Logic, equality (=) is not an application reduces if its operand reduces defined on (abstract) propositions. 1 2 ⇨ (1 2)
1 2 ⇔ ((1 2 2) (1 1 =2)) Direct Logic has the following usual principles for 1 ⇔ ( ( ) ⇨ = ) equality: 1 2 1 1 ⇨ E=E = 1 1 ⇨ ( ) = ⇨ = 1 1 2 1 2 2 1 ⇨ (λ(x) E) (1=2 2=3)⇨ 1=3 E1=E2 ⇨ (1E1 ⇔ 1E2) Nondeterministic λ-calculus (E1= =E2) ⇔ (E1 E2) Direct Logic makes use of the nondeterministic (1=2 1F) ⇨ F(1)=F(2) λ-calculus as follows: (F1=F2 1) ⇨ F1()=F2()
o If E1 and E2 are expressions, then E1 E2 (E1 can P[E] ⇨ (1P 1E) reduce to E2 in the nondeterministic λ-calculus) is a (1=2 1P) ⇨ (P[1] ⇨ P[2]) proposition. 1F ⇨ F=(λ(x) F(x)) abstraction o If E is an expression, then E (E always converges in the nondeterministic λ-calculus) is a proposition. Set Theory o If E is an expression, then E (E is irreducible in The set of all sets in Direct Logic is called Sets and is the nondeterministic λ-calculus) is a proposition. axiomatised below. o If E and E are expressions, then E E (E can 1 2 1 2 1 x: x{ } the empty set { } has no elements converge to E2 in the nondeterministic λ-calculus) is a proposition. sSets: { }s { } is a subset of every set Since Direct Logic uses choice functions instead of o If E is an expression, then 1E (E reduces to exactly 1 expression in the nondeterministic λ- existential quantifiers, we have the following axiom: calculus) is a proposition. sSets: s{ } ⇨ Choice(s)s Note that SetsSets. 113 Reduction roundtripping can be expressed as follows: The basic axioms of set theory are: 114 ├ ( E ⟻ ⟼ E) s1,s2Sets; x: s1s2 ⇨ (xs1 ⇨ xs2) T if s1 is a subset of s2 ,then x is an element Basic axioms are as follows: of s1 implies x is an element of s2
(E = = E) True s1,s2Sets: (s1={ }SubsetChoices2(s1)s2)⇨s1s2 (True = =False) False where (False = =True) False s1,s2Sets: s1≠{ } ⇨ SubsetChoices2(s1)s1
(if True then E1 else E2) E1 if s1 is empty or the choice of an element of s1 (depending in an arbitrary way on s2) is also an element of s2 ,then s1 is a subset of 113 Reduction roundtripping says the reification of has enough s2 information that abstracting back is reduction equivalent to .
114 E1 ⟻ ⟼E2 means that E1 ⟼E2 and E2 ⟼E1
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115 [Clinger 1981; later generalized in Hewitt 2006] answered x; s1,s2Sets: xs1s2 ⇔ (xs1 xs2) the question in the negative by giving a counterexample. x; s1,s2Sets: xs1s2 ⇔ (xs1 xs2) The counterexample is remarkable in that it violates the st x; s1,s2Sets: xs1-s2 ⇔ (xs1 xs2) compactness theorem for 1 order logic: x; y: x{y} ⇔x=y Any finite set of sentences is consistent (the activation order and all arrival orders are discrete) and represents a The function Count is defined as follows: potentially physically realizable situation. But there is Count(s) ≡ an infinite set of sentences that is inconsistent with the discreteness of the combined order and does not if s={ } then 0 else 1+Count(s-{Choice(s)}) represent a physically realizable situation. sSets: Finite[s] ⇔ Count(s) The counterexample is not a problem for Direct Logic a set s is finite if and only if Count(s) converges because the compactness theorem does not hold. The resolution of the problem is to take discreteness of the The integers can be defined as follows using the combined order as an axiom of the Actor model:116 nondeterministic λ-calculus: e1,e2Events: Finite[{eEvents | e1 → e → e2}] IntegerGenerator() ≡ 0 | (1+IntegerGenerator()) IntegerGenerator() is the nondeterministic choice of Direct Logic is based on XML 0 and 1+IntegerGenerator() We speak in strings, but think in trees. 117 ---Nicolaas de Bruijin x: x ⇔ IntegerGenerator()x 118 x is an integer if and only if Integer converges to x The base domain of Direct Logic is XML . In Direct Logic, a dog is an XML dog, e.g., Noncompactness
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121 The following are axioms for XML: Theorem. XML+ is the universe, i.e.,
⇨ (EXML+ EXML+) (Atomics Elements) = XML (Atomics Elements) = { } Atomics and Elements are disjoint Tags Atomics
x: xElements ⇔ x=
A set pXML is defined to be inductive (written Inductive[p] ) if and only it contains the atomics and for all elements that it contains, it also contains every element with those sub-elements :
(pXML; x1…xnp; tTags:
Inductive[p] ⇨ (Atomics p
XML Plus (XML+) is the domain of Direct Logic that is obtained by first extending the Atomics (described above) with Actors119 (see [Hewitt 2009c]) in order to create
XMLwithActors.. Then XML+ is defined recursively by the following axioms:
0 XML+ ≡ XMLwithActors i+1 i i; x: (xXML+ ⇔ xXML+) i XML+ ≡ i XML+
The universe of sets can be defined as follows:120
Sets ≡ XML+ - XMLwithActors
Subsets of elements of XML+ can be defined using the following Restricted Comprehension Axiom:
e: e{x | P[x] } ⇔ (P[e] eXML+)
121 What about Cantor‘s set defined as follows:
Cantor ≡ {xXML+ | xXML+ } Clearly CantorXML+. This illustrates that Cantor is not all
subsets of XML+, just the ones whose elements are in XML+. For 119 λ-expressions are a subset of Actors (see appendix below) example XML+Cantor even though XML+XML+ because
XML XML . It is impossible in Direct Logic to get ―outside‖ 120 + + Note that SetsSets XML+ and its subsets.
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