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Knowledge representation

From Wikipedia, the free encyclopedia

Jump to: navigation, searchKnowledge representation is an area in artificial intelligence that is concerned with how to formally “think”, that is, how to use a symbol system to represent “a domain of discourse” – that which can be talked about, along with functions that may or may not be within the domain of discourse that allow inference (formalized reasoning) about the objects within the domain of discourse to occur. Generally speaking, some kind of logic is used both to supply a formal semantics of how reasoning functions apply to symbols in the domain of discourse, as well as to supply (depending on the particulars of the logic), operators such as quantifiers, modal operators, etc. that, along with an interpretation theory, give meaning to the sentences in the logic.

When we design a knowledge representation (and a knowledge representation system to interpret sentences in the logic in order to derive inferences from them) we have to make trades across a number of design spaces, described in the following sections. The single most important decision to be made, however is the expressivity of the KR. The more expressive, the easier (and more compact) it is to “say something”. However, more expressive languages are harder to automatically derive inferences from. An example of a less expressive KR would be propositional logic. An example of a more expressive KR would be autoepistemic temporal modal logic. Less expressive KRs may be both complete and consistent (formally less expressive than set theory). More expressive KRs may be neither complete nor consistent.

The key problem is to find a KR (and a supporting reasoning system) that can make the inferences your application needs in time, that is, within the resource constraints appropriate to the problem at hand. This tension between the kinds of inferences an application “needs” and what counts as “in time” along with the cost to generate the representation itself makes knowledge representation engineering interesting.

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[edit] Overview

There are representation techniques such as frames, rules and semantic networks which have originated from theories of human information processing. Since knowledge is used to achieve intelligent behavior, the fundamental goal of knowledge representation is to represent knowledge in a manner as to facilitate inferencing (i.e. drawing conclusions) from knowledge.

Some issues that arise in knowledge representation from an AI perspective are:

  • How do people represent knowledge?
  • What is the nature of knowledge and how do we represent it?
  • Should a representation scheme deal with a particular domain or should it be general purpose?
  • How expressive is a representation scheme or formal language?
  • Should the scheme be declarative or procedural?

There has been very little top-down discussion of the knowledge representation (KR) issues and research in this area is a well aged quiltwork. There are well known problems such as “spreading activation” (this is a problem in navigating a network of nodes), “subsumption” (this is concerned with selective inheritance; e.g. an ATV can be thought of as a specialization of a car but it inherits only particular characteristics) and “classification.” For example a tomato could be classified both as a fruit and a vegetable.

In the field of artificial intelligence, problem solving can be simplified by an appropriate choice of knowledge representation. Representing knowledge in some ways makes certain problems easier to solve. For example, it is easier to divide numbers represented in Hindu-Arabic numerals than numbers represented as Roman numerals.

[edit] History of knowledge representation

In computer science, particularly artificial intelligence, a number of representations have been devised to structure information.

KR is most commonly used to refer to representations intended for processing by modern computers, and in particular, for representations consisting of explicit objects (the class of all elephants, or Clyde a certain individual), and of assertions or claims about them (‘Clyde is an elephant’, or ‘all elephants are grey’). Representing knowledge in such explicit form enables computers to draw conclusions from knowledge already stored (‘Clyde is grey’).

Many KR methods were tried in the 1970s and early 1980s, such as heuristic question-answering, neural networks, theorem proving, and expert systems, with varying success. Medical diagnosis (e.g., Mycin) was a major application area, as were games such as chess.

In the 1980s formal computer knowledge representation languages and systems arose. Major projects attempted to encode wide bodies of general knowledge; for example the “Cyc” project (still ongoing) went through a large encyclopedia, encoding not the information itself, but the information a reader would need in order to understand the encyclopedia: naive physics; notions of time, causality, motivation; commonplace objects and classes of objects.

Through such work, the difficulty of KR came to be better appreciated. In computational linguistics, meanwhile, much larger databases of language information were being built, and these, along with great increases in computer speed and capacity, made deeper KR more feasible.

Several programming languages have been developed that are oriented to KR. Prolog developed in 1972,[1] but popularized much later, represents propositions and basic logic, and can derive conclusions from known premises. KL-ONE (1980s) is more specifically aimed at knowledge representation itself. In 1995, the Dublin Core standard of metadata was conceived.

In the electronic document world, languages were being developed to represent the structure of documents, such as SGML (from which HTML descended) and later XML. These facilitated information retrieval and data mining efforts, which have in recent years begun to relate to knowledge representation.

Development of the Semantic Web, has included development of XML-based knowledge representation languages and standards, including RDF, RDF Schema, Topic Maps, DARPA Agent Markup Language (DAML), Ontology Inference Layer (OIL), and Web Ontology Language (OWL).

[edit] Topics in Knowledge representation

[edit] Language and notation

Some people think it would be best to represent knowledge in the same way that it is represented in the human mind, or to represent knowledge in the form of human language.

Psycholinguistics is investigating how the human mind stores and manipulates language. Other branches of cognitive science examine how human memory stores sounds, sights, smells, emotions, procedures, and abstract ideas. Science has not yet completely described the internal mechanisms of the brain to the point where they can simply be replicated by computer programmers.

Various[which?] artificial languages and notations have been proposed for representing knowledge. They are typically based on logic and mathematics, and have easily parsed grammars to ease machine processing. They usually fall into the broad domain of ontologies.

[edit] Ontology languages

Main article: Ontology language

After CycL, a number of ontology languages have been developed. Most are declarative languages, and are either frame languages, or are based on first-order logic. Most of these languages only define an upper ontology with generic concepts, whereas the domain concepts are not part of the language definition. Gellish English is an example of an ontological language that includes a full engineering English Dictionary.

[edit] Links and structures

While hyperlinks have come into widespread use, the closely related semantic link is not yet widely used. The mathematical table has been used since Babylonian times. More recently, these tables have been used to represent the outcomes of logic operations, such as truth tables, which were used to study and model Boolean logic, for example. Spreadsheets are yet another tabular representation of knowledge. Other knowledge representations are trees, by means of which the connections among fundamental concepts and derivative concepts can be shown.

Visual representations are relatively new in the field of knowledge management but give the user a way to visualise how one thought or idea is connected to other ideas enabling the possibility of moving from one thought to another in order to locate required information. The approach is not without its competitors.[2]

[edit] Notation

The recent fashion in knowledge representation languages is to use XML as the low-level syntax. This tends to make the output of these KR languages easy for machines to parse, at the expense of human readability and often space-efficiency.

First-order predicate calculus is commonly used as a mathematical basis for these systems, to avoid excessive complexity. However, even simple systems based on this simple logic can be used to represent data that is well beyond the processing capability of current computer systems: see computability for reasons.

Examples of notations:

[edit] Storage and manipulation

One problem in knowledge representation consists of how to store and manipulate knowledge in an information system in a formal way so that it may be used by mechanisms to accomplish a given task. Examples of applications are expert systems, machine translation systems, computer-aided maintenance systems and information retrieval systems (including database front-ends).

Semantic networks may be used to represent knowledge. Each node represents a concept and arcs are used to define relations between the concepts. One of the most expressive and comprehensively described knowledge representation paradigms along the lines of semantic networks is MultiNet (an acronym for Multilayered Extended Semantic Networks).

From the 1960s, the knowledge frame or just frame has been used. Each frame has its own name and a set of attributes, or slots which contain values; for instance, the frame for house might contain a color slot, number of floors slot, etc.

Using frames for expert systems is an application of object-oriented programming, with inheritance of features described by the “is-a” link. However, there has been no small amount of inconsistency in the usage of the “is-a” link: Ronald J. Brachman wrote a paper titled “What IS-A is and isn’t”, wherein 29 different semantics were found in projects whose knowledge representation schemes involved an “is-a” link. Other links include the “has-part” link.

Frame structures are well-suited for the representation of schematic knowledge and stereotypical cognitive patterns. The elements of such schematic patterns are weighted unequally, attributing higher weights to the more typical elements of a schema. A pattern is activated by certain expectations: If a person sees a big bird, he or she will classify it rather as a sea eagle than a golden eagle, assuming that his or her “sea-scheme” is currently activated and his “land-scheme” is not.

Frame representations are object-centered in the same sense as semantic networks are: All the facts and properties connected with a concept are located in one place – there is no need for costly search processes in the database.

A behavioral script is a type of frame that describes what happens temporally; the usual example given is that of describing going to a restaurant. The steps include waiting to be seated, receiving a menu, ordering, etc. The different solutions can be arranged in a so-called semantic spectrum with respect to their semantic expressivity.

[edit] References

  1. ^ Timeline: A Brief History of Artificial Intelligence, AAAI
  2. ^ Other visual search tools are built by Convera Corporation, Entopia, Inc., EPeople Inc., and Inxight Software Inc.

[edit] Further reading

  • Ronald J. Brachman; What IS-A is and isn’t. An Analysis of Taxonomic Links in Semantic Networks; IEEE Computer, 16 (10); October 1983 [1]
  • Ronald J. Brachman, Hector J. Levesque Knowledge Representation and Reasoning, Morgan Kaufmann, 2004 ISBN 978-1-55860-932-7
  • Ronald J. Brachman, Hector J. Levesque (eds) Readings in Knowledge Representation, Morgan Kaufmann, 1985, ISBN 0-934613-01-X
  • Randall Davis, Howard Shrobe, and Peter Szolovits; What Is a Knowledge Representation? AI Magazine, 14(1):17-33,1993 [2]
  • Ronald Fagin,Joseph Y. Halpern,Yoram Moses,Moshe Y. Vardi Reasoning About Knowledge, MIT Press, 1995, ISBN 0-262-06162-7
  • Jean-Luc Hainaut, Jean-Marc Hick, Vincent Englebert, Jean Henrard, Didier Roland: Understanding Implementations of IS-A Relations. ER 1996: 42-57 [3]
  • Hermann Helbig: Knowledge Representation and the Semantics of Natural Language, Springer, Berlin, Heidelberg, New York 2006
  • Arthur B. Markman: Knowledge Representation Lawrence Erlbaum Associates, 1998
  • John F. Sowa: Knowledge Representation: Logical, Philosophical, and Computational Foundations. Brooks/Cole: New York, 2000
  • Adrian Walker, Michael McCord, John F. Sowa, and Walter G. Wilson: Knowledge Systems and Prolog, Second Edition, Addison-Wesley, 1990

[edit] See also

see also: Category:Knowledge representation

 

 
 

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Dr. Know (A.I. Artificial Intelligence) • The Librarian (Snow Crash)

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http://users.ox.ac.uk/~jrlucas/Godel/brighton.html

“Minds, Machines and Gödel: A Retrospect”, in P.J.R.Millican and A.Clark, eds., Machines and Thought: The Legacy of Alan Turing, Oxford, 1996, pp.103-124.
A paper read to the Turing Conference at Brighton on April 6th, 1990
by
J.R. Lucas
Fellow of Merton College, Oxford
I must start with an apologia. My original paper, “Minds, Machines and Gödel”, was written in the wake of Turing’s 1950 paper in Mind, and was intended to show that minds were not Turing machines. Why, then, didn’t I couch the argument in terms of Turing’s theorem, which is easyish to prove and applies directly to Turing machines, instead of Gödel’s theorem, which is horrendously difficult to prove, and doesn’t so naturally or obviously apply to machines? The reason was that Gödel’s theorem gave me something more: it raises questions of truth which evidently bear on the nature of mind, whereas Turing’s theorem does not; it shows not only that the Gödelian well-formed formula is unprovable-in-the-system, but that it is true. It shows something about reasoning, that it is not completely rule-bound, so that we, who are rational, can transcend the rules of any particular logistic system, and construe the Gödelian well-formed formula not just as a string of symbols but as a proposition which is true. Turing’s theorem might well be applied to a computer which someone claimed to represent a human mind, but it is not so obvious that what the computer could not do, the mind could. But it is very obvious that we have a concept of truth. Even if, as was claimed in a previous paper, it is not the summum bonum, it is a bonum, and one it is characteristic of minds to value. A representation of the human mind which could take no account of truth would be inherently implausible. Turing’s theorem, though making the same negative point as Gödel’s theorem, that some things cannot be done by even idealised computers, does not make the further positive point that we, in as much as we are rational agents, can do that very thing that the computer cannot. I have however, sometimes wondered whether I could not construct a parallel argument based on Turing’s theorem, and have toyed with the idea of a von Neumann machine. A von Neumann machine was a black box, inside which was housed John von Neumann. But although it was reasonable, on inductive grounds, to credit a von Neumann machine with the power of solving any problem in finite time—about the time taken to get from New York to Chicago by train—it did not have the same edge as Gödel’s proof of his own First Incompleteness Theorem. I leave it therefore to members of this conference to consider further how Turing’s theorem bears on mechanism, and whether a Turing machine could plausibly represent a mind, and return to the argument I actually put forward.
I argued that Gödel’s theorem enabled us to devise a schema for refuting the various different mechanist theories of the mind that might be put forward. Gödel’s theorem is a sophisticated form of the Cretan paradox posed by Epimenides. Gödel showed how we could represent any reasonable mathematical theory within itself. Whereas the original Cretan paradox, `This statement is untrue’ can be brushed off on the grounds that it is viciously self-referential, and we do not know what the statement is, which is alleged to be untrue, until it has been made, and we cannot make it until we know what it is that is being alleged to be false, Gödel blocks that objection. But in order to do so, he needs not only to represent within his mathematical theory some means of referring to the statement, but also some means of expressing mathematically what we are saying about it. We cannot in fact do this with `true’ or `untrue’: could we do that, a direct inconsistency would ensue. What Gödel was able to do, however, was to express within his mathematical system the concept of being provable-, and hence also unprovable-, in-that-system. He produced a copper-bottomed well-formed formula which could be interpreted as saying `This well-formed formula is unprovable-in-this-system’. It follows that it must be both unprovable-in-the-system and none the less true. For if it were provable, and provided the system is a sound one in which only well-formed formulae expressing true propositions could be proved, then it would be true, and so what it says, namely that it is unprovable-in-the-system, would hold; so that it would be unprovable-in-the-system. So it cannot be provable-in-the-system. But if it is unprovable-in-the-system, then what it claims to be the case is the case, and so it is true. So it is true but unprovable-in-the-system. Gödel’s theorem seemed to me to be not only a surprising result in mathematics, but to have a bearing on theories of the mind, and in particular on mechanism, which, as Professor Clark Glymour pointed out two days ago, is as much a background assumption of our age as classical materialism was towards the end of the last century in the form expressed by Tyndale. Mechanism claims that the workings of the mind can be entirely understood in terms of the working of a definite finite system operating according to definite deterministic laws. Enthusiasts for Artificial Intelligence are often mechanists, and are inclined to claim that in due course they will be able to simulate all forms of intelligent behaviour by means of a sufficiently complex computer garbed in sufficiently sophisticated software. But the operations of any such computer could be represented in terms of a formal logistic calculus with a definite finite number (though enormously large) of possible well-formed formulae and a definite finite number (though presumably smaller) of axioms and rules of inference. The Gödelian formula of such a system would be one that the computer, together with its software, would be unable to prove. We, however, could. So the claim that a computer could in principle simulate all our behaviour breaks down at this one, vital point.
The argument I put forward is a two-level one. I do not offer a simple knock-down proof that minds are inherently better than machines, but a schema for constructing a disproof of any plausible mechanist thesis that might be proposed. The disproof depends on the particular mechanist thesis being maintained, and does not claim to show that the mind is uniformly better than the purported mechanist representation of it, but only that it is one respect better and therefore different. That is enough to refute that particular mechanist thesis. By itself, of course, it leaves all others unrefuted, and the mechanist free to put forward some variant thesis which the counter-argument I constructed does not immediately apply to. But I claim that it can be adjusted to meet the new variant. Having once got the hang of the Gödelian argument, the mind can adapt it appropriately to meet each and every variant claim that the mind is essentially some form of Turing machine. Essentially, therefore, the two parts of my argument are first a hard negative argument, addressed to a mechanist putting forward a particular claim, and proving to him, by means he must acknowledge to be valid, that his claim is untenable, and secondly a hand-waving positive argument, addressed to intelligent men, bystanders as well as mechanists espousing particular versions of mechanism, to the effect that some sort of argument on these lines can always be found to deal with any further version of mechanism that may be thought up.
I read the paper to the Oxford Philosophical Society in October 1959 and subsequently published it in Philosophy, 1 and later set out the argument in more detail in The Freedom of the Will. 2 I have been much attacked. Although I argued with what I hope was becoming modesty and a certain degree of tentativeness, many of the replies have been lacking in either courtesy or caution. I must have touched a raw nerve. That, of course, does not prove that I was right. Indeed, I should at once concede that I am very likely not to be entirely right, and that others will be able to articulate the arguments more clearly, and thus more cogently, than I did. But I am increasingly persuaded that I was not entirely wrong, by reason of the very wide disagreement among my critics about where exactly my arguments fail. Each picks on a different point, allowing that the points objected to by other critics, are in fact all right, but hoping that his one point will prove fatal. None has, so far as I can see. I used to try and answer each point fairly and fully, but the flesh has grown weak. Often I was simply pointing out that the critic was not criticizing any argument I had put forward but one which he would have liked me to put forward even though I had been at pains to discount it. In recent years I have been less zealous to defend myself, and often miss articles altogether. 3 There may be some new decisive objection I have altogether overlooked. But the objections I have come across so far seem far from decisive.
To consider each objection individually would be too lengthy a task to attempt here. I shall pick on five recurrent themes. Some of the objections question the idealisation implicit in the way I set up the contest between the mind and the machine; some raise questions of modality and finitude; some turn on issues of transfinite arithmetic; some are concerned with the extent to which rational inferences should be formalisable; and some are about consistency.
Many philosophers question the idealisation implicit in the Gödelian argument. A context is envisaged between “the mind” and “the machine”, but it is an idealised mind and an idealised machine. Actual minds are embodied in mortal clay; actual machines often malfunction or wear out. Since actual machines are not Turing machines, not having an infinite tape, that is to say an infinite memory, it may be held that they cannot be automatically subject to Gödelian limitations. But Gödel’s theorem applies not only to Peano Arithmetic, with its infinitistic postulate of recursive reasoning, but to the weaker Robinson Arithmetic Q, which is only potentially, not actually infinite, and hardly extends beyond the range of plausible computer progress. In any case, limitations of finitude reduce, rather than enhance, the plausibility of some computer’s being an adequate representation of a mind. Actual minds are embodied in mortal clay. In the short span of our actual lives we cannot achieve all that much, and might well have neither the time nor the cleverness to work out our Gödelian formula. Hanson points out that there could be a theorem of Elementary Number Theory that I cannot prove because a proof of it would be too long or complex for me to produce. 4 Any machine that represented a mind would be would be enormously complicated, and the calculation of its Gödel sentence might well be beyond the power of any human mathematician. 5 But he could be helped. Other mathematicians might come to his aid, reckoning that they also had an interest in the discomfiture of the mechanical Goliath. 6 The truth of the Gödelian sentence under its intended interpretation in ordinary informal arithmetic is a mathematical truth, which even if pointed out by other mathematicians would not depend on their testimony in the way contingent statements do. So even if aided by the hints of other mathematicians, the mind’s asserting the truth of the Gödelian sentence would be a genuine ground for differentiating it from the machine.
Some critics of the Gödelian argument—Dennett, Hofstadter and Kirk—complain that I am insufficiently sensitive to the sophistication of modern computer technology, and that there is a fatal ambiguity between the fundamental level of the machine’s operations and the level of input and output that is supposed to represent the mind: in modern parlance, between the machine code and the programming language, such as PROLOG. But although there is a difference of levels, it does not invalidate the argument. A compiler is entirely deterministic. Any sequence of operations specified in machine code can be uniquely specified in the programming language, and vice versa. Hence it is quite fair to characterize the capacity of the mechanist’s machine in terms of a higher level language. In order to begin to be a representation of a mind it must be able to do simple arithmetic. And then, at this level, Gödel’s theorem applies. The same counter applies to Dennett’s complaint that the comparison between men and Turing machines is highly counterintuitive because we are not much given to wandering round uttering obscure truths of ordinary informal arithmetic. Few of us are capable of asserting a Gödelian sentence, fewer still of wanting to do so. “Men do not sit around uttering theorems in a uniform vocabulary, but say things in earnest and in jest, make slips of the tongue, speak several languages, signal agreement by nodding or otherwise acting non-verbally, and—most troublesome for this account—utter all kinds of nonsense and contradictions, both deliberately and inadvertently.” 7 Of course, men are un-machinelike in these ways, and many philosophers have rejected the claims of mechanism on these grounds alone. But mechanists claim that this is too quick. Man, they say, is a very complicated machine, so complicated as to produce all this un-machinelike output. We may regard their contention as highly counter-intuitive, but should not reject it out of hand. I therefore take seriously, though only in order to refute it, the claim that a machine could be constructed to represent the behaviour of a man. If so, it must, among other things, represent a man’s mental behaviour. Some men, many men, are capable of recognising a number of basic arithmetical truths, and, particularly when asked to (which can be viewed as a particular input), can assert them as truths. Although “a characterization of a man as a certain sort of theorem-proving machine” 8 would be a less than complete characterization, it would be an essential part of a characterization of a machine if it was really to represent a man. It would have to be able to include in its output of what could be taken as assertions the basic truths of arithmetic, and to accept as valid inferences those that are validated by first-order logic. This is a minimum. Of course it may be able to do much more—it may have in its memory a store of jokes for use in after-dinner speeches, or personal reminiscences for use on subordinates – but unless its output, for suitable questions or other input, includes a set of assertions itself including Elementary Number Theory, it is a poor representation of some human minds. If it cannot pass O-level maths, are we really going to believe a mechanist when he claims that it represents a graduate?
Actual minds are finite in what they actually achieve. Wang and Boyer see difficulties in the infinite capabilities claimed for the mind as contrasted with the actual finitude of human life. Boyer takes a post mortem view, and points out that all of the actual output of Lucas, Astaire, or anyone else can be represented ex post facto by a machine. 9 Actual achievements of mortal men are finite, and so simulable. When I am dead it would be possible to program a computer with sufficient graphic capacity to show on a video screen a complete biographical film of my life. But when I am dead it will be easy to outwit me. What is in issue is whether a computer can copy a living me, when I have not as yet done all that I shall do, and can do many different things. It is a question of potentiality rather than actuality that is in issue. Wang concedes this, and allows that we are inclined to say that it is logically possible to have a mind capable of recognising any true proposition of number theory or solving a set of Turing-unsolvable problems, but life is short. 10 In a finite life-span only a finite number of the propositions can be recognised, only a finite set of problems can be solved. And a machine can be programmed to do that. Of course, we reckon that a man can go on to do more, but it is difficult to capture that sense of infinite potentiality. This is true. It is difficult to capture the sense of infinite potentiality. But it is an essential part of the our concept of mind, and a modally “flat” account of the a mind in terms only of what it has done is as unconvincing as an account of cause which considers only constant conjunction, and not what would have been the case had circumstances been different. In order to capture this sense of potentiality, I set out my argument in terms of a challenge which leaves it open to the challenger to meet in any way he likes. Two-sided, or “dialectical”, arguments often succeed in encapsulating concepts that elude explication in purely monologous terms: the epsilon-delta exegesis an infinitesimals is best conveyed thus, and more generally any alternation of quantifiers, as in the EA principles suggested by Professor Clark Glymour for the ultimate convergence of theories on truth.
Although some degree of idealisation seems allowable in considering a mind untrammelled by mortality and a Turing machine with infinite tape, doubts remain as to how far into the infinite it is permissible to stray. Transfinite arithmetic underlies the objections of Good and Hofstadter. The problem arises from the way the contest between the mind and the machine is set up. The object of the contest is not to prove the mind better than the machine, but only different from it, and this is done by the mind’s Gödelizing the machine. It is very natural for the mechanist to respond by including the Gödelian sentence in the machine, but of course that makes the machine a different machine with a different Gödelian sentence all of its own, which it cannot produce as true but the mind can. So then the mechanist tries adding a Gödelizing operator, which gives, in effect a whole denumerable infinity of Gödelian sentences. But this, too, can be trumped by the mind, who produces the Gödelian sentence of the new machine incorporating the Gödelizing operator, and out Gödelizes the lot. Essentially this is the move from w (omega), the infinite sequence of Gödelian sentences produced by the Gödelizing operator, to w + 1, the next transfinite ordinal. And so it goes on. Every now and again the mechanist loses patience, and incorporates in his machine a further operator, designed to produce in one fell swoop all the Gödelian sentences the mentalist is trumping him with: this is in effect to produce a new limit ordinal. But such ordinals, although they have no predecessors, have successors just like any other ordinal, and the mind can out-Gödel them by producing the Gödelian sentence of the new version of the machine, and seeing it to be true, which the machine cannot. Hofstadter thinks there is a problem for the mentalist in view of a theorem of Church and Kleene on Formal Definitions of Transfinite Ordinals. 11 They showed that we couldn’t program a machine to produce names for all the ordinal numbers. Every now and again some new, creative step is called for, when we consider all the ordinal numbers hitherto named, and we need to encompass them all in a single set, which we can use to define a new sort of ordinal, transcending all previous ones. Hofstadter thinks that, in view of the Church-Kleene theorem, the mind might run out of steam, and fail to think up new ordinals as required, and so fail in the last resort to establish the mind’s difference from some machine. But this is wrong on two counts. In the first place it begs the question and in the second it misconstrues the nature of the contest.
Hofstadter assumes that the mind is subject to the same limitations as the machine is, and that since there is no mechanical way of naming all the ordinals, the mind cannot do it either. But this is precisely the point in issue. Gödel himself rejected mechanism on account of our ability to think up fresh definitions for transfinite ordinals (and ever stronger axioms for set theory) and Wang is inclined to do so too. 12 On this occasion, it is pertinent to note that Turing himself was, on this question, of the same mind as Gödel. He was led “to ordinal logics as a way to `escape’ Gödel’s incompleteness theorems”, 13 but recognised that “although in pre-Gödel times it was thought by some that it would be able to carry this programmme to such an extent that … the necessity for intuition would be entirely eliminated,” as a result of Gödel’s incompleteness theorems one must turn instead to `non-constructive’ systems of logic in which “not all the steps in a proof are mechanical, some being intuitive”. Turing concedes that the steps whereby we recognise formulae as ordinal formulae are intuitive, and goes on to say that we should show quite clearly when a step makes use of intuition, and when it is purely formal, and that the strain put on intuition should be a minimum. 14 He clearly, like Gödel, allows that the mind’s ability to recognise new ordinals outruns the ability of any formal algorithm to do so, though he does not draw Gödel’s conclusion. It may be, indeed, that the mind’s ability to recognise new ordinals is the issue on which battle should be joined; Good claimed as much 15—though disputes about the notation for ordinals lack the sharp edge of the Gödelian argument. But whatever the merits of different battlefields, it is clear that they are contested areas in the same conflict, and undisputed possession of the one cannot be claimed in order to assert possession of the other.
In any case Hofstadter misconstrues the nature of the contest. All the difficulties are on the side of the mechanist trying to devise a machine that cannot be out-Gödelized. It is the mechanist who resorts to limit ordinals, and who may have problems in devising new notations for them. The mind needs only to go on to the next one, which is always an easy, unproblematic step, and out-Gödelize whatever is the mechanist’s latest offering. Hofstadter’s argument, as often, tells against the position he is arguing for, and shows up a weakness of machines: there is no reason to suppose that it is shared by minds, and in the nature of the case it is a difficulty for those who are seeking to evade the Gödelian argument, not those who are deploying it.
Underlying Hofstadter’s argument is a rhetorical question that many mechanists have raised. “How does Lucas know that the mind can do this, that, or the other?” It is no good, they hold, that I should opine it or simply assert it; I must prove it. And if I prove it, then since the steps of my proof can be programmed into a machine, the machine can do it too. Good puts the argument explicitly:
What he must prove is that he personally can always make the improvement: it is not sufficient to believe it since belief is a matter of probability and Turing machines are not supposed to be capable of probability judgements. But no such proof is possible since, if it were given, it could be used for the design of a machine that could always do the improving.
The same point is made by Webb in his sustained and searching critique of the Gödelian argument:
It is only because Gödel gives an effective way of constructing the Gödelian sentence that Lucas can feel confident that he can find the Achilles’ heel of any machine. But then if Lucas can effectively stump any machine, then there must be a machine which does this too. 16 [This] “is the basic dilemma confronting anti-mechanism: just when the constructions used in its arguments become effective enough to be sure of, (T) <viz. Every humanly effective computation procedure can be simulated by a Turing machine> then implies that a machine can simulate them. In particular it implies that our very behaviour of applying Gödel’s argument to arbitrary machines – in order to conclude that we cannot be modelled by a machine – can indeed be modelled by a machine. Hence any such conclusion must fail, or else we will have to conclude that certain machines cannot be modelled by any machine! In short, anti-mechanist arguments must either be ineffective, or else unable to show that their executor is not a machine.” 17
The core of this argument is an assumption that every informal argument must either be formalisable or else invalid. Such an assumption undercuts the distinction I have drawn between two senses of Gödelian argument: between a negative argument according to an exact specification, which a machine could be programmed to carry out, and on the other hand a certain style of arguing, similar to Gödel’s original argument in inspiration, but not completely or precisely specified, and therefore not capable of being programmed into a machine, though capable of being understood and applied by an intelligent mind. Admittedly, we cannot prove to a hide-bound mechanist that we can go on. But we may come to a well-grounded confidence that we can, which will give us, and the erstwhile mechanist if he is reasonable and not hide-bound, good reason for rejecting mechanism.
Against this claim of the mentalist that he has got the hang of doing something which cannot be described in terms of a mechanical program, the mechanist says “Sez you” and will not believe him unless he produces a program showing how he would do it. It is like the argument between the realist and the phenomenalist. The realist claims that there exist entities not observed by anyone: the phenomenalist demands empirical evidence; if it is not forthcoming, he remains sceptical of the realist’s claim; if it is, then the entity is not unobserved. In like manner the mechanist is sceptical of the mentalist’s claim unless he produces a specification of how he would do what a machine cannot: if such a specification is not forthcoming, he remains sceptical; if it is, it serves as a basis for programming a machine to do it after all. The mechanist position, like the phenomenalist, is invulnerable but unconvincing. I cannot prove to the mechanist that anything can be done other than what a machine can do, because he has restricted what he will accept as a proof to such an extent that only “machine-doable” deeds will be accounted doable at all. But not all mechanists are so limited. Many mechanists and many mentalists are rational agents wondering whether in the light of modern science and cybernetics mechanism is, or is not, true. They have not closed their minds by so redefining proof that none but mechanist conclusions can be established. They can recognise in themselves their having “got the hang” of something, even though no program can be written for giving a machine the hang of it. The parallel with the Sorites argument is helpful. Arguing against a finitist, who does not accept the principle of mathematical induction, I may see at the meta-level that if he has conceded F(0) and (Ax)(F(x) –> F(x + 1)) then I can claim without fear of contradiction (Ax)F(x). I can be quite confident of this, although I have no finitist proof of it. All I can do, vis à vis the finitist, is to point out that if he were to deny my claim in any specific instance, I could refute him. True, a finitist could refute him too. But I have generalised in a way a finitist could not, so that although each particular refuting argument is finite, the claim is infinite. In a similar fashion each Gödelian argument is effective, and will convince even the mechanist that he is wrong; but the generalisation from individual tactical refutations to a strategic claim does not have to be effective in the same sense, although it may be entirely rational for the mind to make the claim.
Nevertheless an air of paradox remains. The idea of a totally intuitive, unformalisable argument arouses suspicion: if it can convince, it can be conveyed, and if it can be conveyed, it can be formulated and expressed in formal terms. Let me therefore stress that I am not claiming that my, or any, argument is absolutely unformalisable. Any argument can be formalised, as the Tortoise proved to Achilles, the formal axiom or rule of inference invoked will be no more convincing than the original unformalised argument. I am not claiming that the Gödelian argument cannot be formalised, but that whatever formalisation we adopt, there are further arguments which are clearly valid but not captured by that formalisation. Not only, again as the Tortoise proved to Achilles, must we always be ready to recognise some rules of inference as applying and inferences as valid without more ado, but we shall be led, if we are rational, to extend our range of acknowledged valid inferences beyond any antecedently laid down bounds. This does not preclude our subsequently formalising them, only our supposing that any formalisation is inferentially complete.
But we always can formalise; in particular, we can formalise the argument that Gödel uses to prove that the Gödelian formula is unprovable-in-the-system but none the less true. At first sight there seems to be a paradox. Gödel’s argument purports to show that the Gödelian sentence is unprovable but true. But if it shows that the Gödelian sentence is true, surely it has proved it, so that it is provable after all. The paradox in this case is resolved by distinguishing provability-in-the-formal-system from the informal provability given by Gödel’s reasoning. But this reasoning can be formalised. We can go over Gödel’s argument step by step, and formalise it. If we do so we find that an essential assumption for his argument that the Gödelian sentence is unprovable is that the formal system should be consistent. Else every sentence would be provable, and the Gödelian sentence, instead of being unprovable and therefore true, could be provable and false. So what we obtain, if we formalise Gödel’s informal argumentation, is not a formal proof within Elementary Number Theory (ENT for short) that the Gödelian sentence, G is true, but a formal proof within Elementary Number Theory
|- Cons(ENT) –> G
where Cons(ENT) is a sentence expressing the consistency of Elementary Number Theory. Only if we also had a proof in Elementary Number Theory yielding
|- Cons(ENT)
would we be able to infer by Modus Ponens
|- G
Since we know that
¬ |- G, [i.e. G is not derivable: this is the best I can do to render symbolic logic in HTML]
we infer also that
¬:|- Cons(ENT). [i.e. Cons(ENT) is not derivable]
This is Gödel’s second theorem. Many critics have appealed to it in order to fault the Gödelian argument. Only if the machine’s formal system is consistent and we are in a position to assert its consistency are we really able to maintain that the Gödelian sentence is true. But we have no warrant for this. For all we know, the machine we are dealing with may be inconsistent, and even if it is consistent we are not entitled to claim that it is. And in default of such entitlement, all we have succeeded in proving is
|- Cons(ENT) –> G,
and the machine can do that too.
These criticisms rest upon two substantial points: the consistency of the machine’s system is assumed by the Gödelian argument and cannot be always established by a standard decision-procedure. The question “By what right does the mind assume that the machine is consistent?” is therefore pertinent. But the moves made by mechanists to deny the mind that knowledge are unconvincing. Paul Benacerraf suggests that the mechanist can escape the Gödelian argument by not staking out his claim in detail. 18 The mechanist offers a “Black Box” without specifying its program, and refusing to give away further details beyond the claim that the black box represents a mind. But such a position is both vacuous and untenable: vacuous because there is no content to mechanism unless some specification is given—if I am presented with a black box but “told not to peek inside” then why should I think it contains a machine and not, say, a little black man? The mechanist’s position is also untenable: for although the mechanist has refused to specify what machine it is that he claims to represent the mind, it is evident that the Gödelian argument would work for any consistent machine and that an inconsistent machine would be an implausible representation. The stratagem of playing with his cards very close to his chest in order to deny the mind the premisses it needs is a confession of defeat.
Putnam contends that there is an illegitimate inference from the true premiss
I can see that (Cons(ENT) —> G)
to the false conclusion
Cons(ENT) –> I can see that (G). 19
It is the latter that is needed to differentiate the mind from the machine, for what Gödel’s theorem shows is
Cons(ENT) —> ENT machine cannot see that (G),
but it is only the former, according to Putnam, that I am entitled to assert. Putnam’s objection fails on account of the dialectical nature of the Gödelian argument. The mind does not go round uttering theorems in the hope of tripping up any machines that may be around. Rather, there is a claim being seriously maintained by the mechanist that the mind can be represented by some machine. Before wasting time on the mechanist’s claim, it is reasonable to ask him some questions about his machine to see whether his seriously maintained claim has serious backing. It is reasonable to ask him not only what the specification of the machine is, but whether it is consistent. Unless it is consistent, the claim will not get off the ground. If it is warranted to be consistent, then that gives the mind the premiss it needs. The consistency of the machine is established not by the mathematical ability of the mind but on the word of the mechanist. The mechanist has claimed that his machine is consistent. If so, it cannot prove its Gödelian sentence, which the mind can none the less see to be true: if not, it is out of court anyhow.
Wang concedes that it is reasonable to contend that only consistent machines are serious candidates for representing the mind, but then objects it is too stringent a requirement for the mechanist to meet because there is no decision-procedure that will always tell us whether a formal system strong enough to include Elementary Number Theory is consistent or not. 20 But the fact that there is no decision-procedure means only that we cannot always tell, not that we can never tell. Often we can tell that a formal system is not consistent—e.g. it proves as a theorem:
|- p&¬p
or,
|- 0 = 1
Also, we may be able to tell that a system is consistent. We have finitary consistency proofs for propositional calculus and first-order predicate calculus, and Gentzen’s proof, involving transfinite induction, for Elementary Number Theory. We are therefore not asking the impossible of the mechanist in requiring him to do some preliminary sorting out before presenting candidates for being plausible representations of the mind. Unless they satisfy the examiner—the mechanist—in Prelims on the score of consistency, they are not eligible to enter for Finals, and all those that are thus qualified can be sure of failing for not being able to assert their Gödelian sentence.
The two-stage examination is thus able to sort out the inconsistent sheep who fail the qualifying examination from the consistent goats who fail their finals, and hence enables us to take on all challenges even from inconsistent machines, without pretending to possess superhuman powers. Although all machines are entitled to enter for the mind-representation examination, only relatively few machines are plausible candidates for representing the mind, and there is no need to take a candidate seriously just because it is a machine. If the mechanist’s claim is to be taken seriously, some recommendation will be required, and at the very least a warranty of consistency would be essential. Wang protests that this is to expect superhuman powers of him, and in a response to Benacerraf’s “God, The Devil and Gödel”, I picked up his suggestion that the mechanist might be no mere man but the Prince of Darkness himself to whom the question of whether the machine was consistent or not could be addressed in expectation of an answer. 21 Rather than ask high-flown questions about the mind we can ask the mechanist the single question whether or not the machine that is proposed as a representation of the mind would affirm the Gödelian sentence of its system. If the mechanist says that his machine will affirm the Gödelian sentence, the mind then will know that it is inconsistent and will affirm anything, quite unlike the mind which is characteristically selective in its intellectual output. If the mechanist says that his machine will not affirm the Gödelian sentence, the mind then will know since there was at least one sentence it could not prove in its system it must be consistent; and knowing that, the mind will know that the machine’s Gödelian sentence is true, and thus will differ from the machine in its intellectual output. And if the mechanist is merely human, and moreover does not know what answer the machine would give to the Gödelian question, he has not done his home-work properly, and should go away and try to find out before expecting us to take him seriously.
In asking the mechanist rather than the machine, we are making use of the fact that the issue is one of principle, not of practice. The mechanist is not putting forward actual machines which actually represent some human being’s intellectual output, but is claiming instead that there could in principle be such a machine. He is inviting us to make an intellectual leap, extrapolating from various scientific theories and skating over many difficulties. He is quite entitled to do this. But having done this he is not entitled to be coy about his in-principle machine’s intellectual capabilities or to refuse to answer embarrassing questions. The thought-experiment, once undertaken, must be thought through. And when it is thought through it is impaled on the horns of a dilemma. Either the machine can prove in its system the Gödelian sentence or it cannot: if it can, it is inconsistent, and not equivalent to a mind; if it cannot, it is consistent, and the mind can therefore assert the Gödelian sentence to be true. Either way the machine is not equivalent to the mind, and the mechanist thesis fails.
A number of thinkers have chosen to impale themselves on the inconsistency horn of the dilemma. We are machines, they say, but very limited, fallible and inconsistent ones. In view of our many contradictions, changes of mind and failures of logic, we have no warrant for supposing the mind to be consistent, and therefore no ground for disqualifying a machine for inconsistency as a candidate for being a representation of the mind. Hofstadter thinks it would be perfectly possible to have an artificial intelligence in which propositional reasoning emerged as consequences rather than as being pre-programmed. “And there is no particular reason to assume that the strict Propositional Calculus, with its rigid rules and the rather silly definition of consistency they entail, would emerge from such a program.” 22
None of these arguments goes any way to making an inconsistent machine a plausible representation of a mind. Admittedly the word `consistent’ is used in different senses, and the claim that a mind is consistent is likely to involve a different sense of consistency and to be established by different sorts of arguments from those in issue when a machine is said to be consistent. If this is enough to establish the difference between minds and machines, well and good. But many mechanists will not be so quickly persuaded and will maintain that a machine can be programmed, in some such way as Hofstadter supposes, to emit mind-like behaviour. In that case it is machine-like consistency rather than mind-like consistency that is in issue. Any machine, if it is to begin to represent the output of a mind must be able to operate with symbols that can be plausibly interpreted as negation, conjunction, implication, etc., and so must be subject to the rules of some variant of the propositional calculus. Unless something rather like the propositional calculus with some comparable requirement of consistency emerges from the program of a machine, it will not be a plausible representation of a mind, no matter no matter how good it is as a specimen of Artificial Intelligence. Of course, any plausible representation of a mind would have to manifest the behaviour instanced by Wang, constantly checking whether a contradiction had been reached and attempting to revise its basic axioms when that happened. But this would have to be in accordance with certain rules. There would have to be a program giving precise instructions how the checking was to be undertaken, and in what order axioms were to be revised. Some axioms would need to be fairly immune to revision. Although some thinkers are prepared to envisage a logistic calculus in which the basic inferences of propositional calculus do not hold (e.g. from p & q to p) or the axioms of Elementary Number Theory have been rejected, any machine which resorted to such a stratagem to avoid contradiction would also lose all credence as a representation of a mind. Although we sometimes contradict ourselves and change our minds, some parts of our conceptual structure are very stable, and immune to revision. Of course it is not an absolute immunity. One can allow the Cartesian possibility of conceptual revision without being guilty, as Hutton supposes, 23 of inconsistency in claiming knowledge of his own consistency. To claim to know something is not to claim infallibility but only to have adequate backing for what is asserted. Else all knowledge of contingent truths would be impossible. Although one cannot say `I know it, although I may be wrong’, it is perfectly permissible to say `I know it, although I might conceivably be wrong’. So long as a man has good reasons, he can responsibly issue a warranty in the form of a statement that he knows, even though we can conceive of circumstances in which his claim would prove false and would have to be withdrawn. So it is with our claim to know the basic parts of our conceptual structure, such as the principles of reasoning embodied in the propositional calculus or the truths of ordinary informal arithmetic. We have adequate, more than adequate, reason for affirming our own consistency and the truth, and hence also the consistency, of informal arithmetic, and so can properly say that we know, and that any machine representation of the mind must manifest an output expressed by a formal (since it is a machine) system which is consistent and includes Elementary Number Theory (since it is supposed to represent the mind). But there remains the Cartesian possibility of our being wrong, and that we need now to discuss. Some mechanists have conceded that a consistent machine could be out-Gödeled by a mind, but have maintained that the machine representation of the mind is an inconsistent machine, but one whose inconsistency is so deep that it would take a long time ever to come to light. It therefore would avoid the quick death of non-selectivity. Although in principle it could be brought to affirm anything, in practice it will be selective, affirming some things and denying others. Only in the long run will it age—or mellow, as we kindly term it—and then “crash” and cease to deny anything; and in the long run we die—usually before suffering senile dementia. Such a suggestion chimes in with a line of reasoning which has been noticeable in Western Thought since the Eighteenth Century. Reason, it is held, suffers from certain antinomies, and by its own dialectic gives rise to internal contradictions which it is quite powerless to reconcile, and which must in the end bring the whole edifice crashing down in ruins. If the mind is really an inconsistent machine then the philosophers in the Hegelian tradition who have spoken of the self-destructiveness of reason are simply those in whom the inconsistency has surfaced relatively rapidly. They are the ones who have understood the inherent inconsistency of reason, and who, negating negation, have abandoned hope of rational discourse, and having brought mind to the end of its tether, have had on offer only counsels of despair.
Against this position the Gödelian argument can avail us nothing. Quite other arguments and other attitudes are required as antidotes to nihilism. It has long been sensed that materialism leads to nihilism, and the Gödelian argument can be seen as making this reductio explicit. And it is a reductio. For mechanism claims to be a rational position. It rests its case on the advances of science, the underlying assumptions of scientific thinking and the actual achievements of scientific research. Although other people may be led to nihilism by feelings of angst or other intimations of nothingness, the mechanist must advance arguments or abandon his advocacy altogether. On the face of it we are not machines. Arguments may be adduced to show that appearances are deceptive, and that really we are machines, but arguments presuppose rationality, and if, thanks to the Gödelian argument, the only tenable form of mechanism is that we are inconsistent machines, with all minds being ultimately inconsistent, then mechanism itself is committed to the irrationality of argument, and no rational case for it can be sustained.
 

 

To return from footnote to text, click on footnote number
1. “ Minds, Machines and Gödel” Philosophy, 36, 1961, pp.112-127; reprinted in Kenneth M.Sayre and Frederick J.Crosson, eds., The Modeling of Mind, Notre Dame, 1963, pp. 255-271; and in A.R.Anderson, Minds and Machines, Prentice-Hall, 1964, pp. 43-59.
2. The Freedom of the Will, Oxford, 1970 (now avaialble again).
3. I give at the end a list of some of the major criticisms I have come across.
4. William Hanson, “Mechanism and G”del’s Theorems,” British Journal for the Philosophy of Science, XXII, 1971, p.12; compare Hofstadter, 1979, p.475.
5. Rudy Rucker, “G”del’s Theorem: The Paradox at the heart of modern man”, Popular Computing, February 1985, p.168.
6. I owe this suggestion to M.A.E. Dummett, at the original meeting of the Oxford Philosophical Society on October 30th, 1959. A similar suggestion is implicit in Hao Wang, From Mathematics to Philosophy, London, 1974, p.316.
7. D.C.Dennett, Review of The Freedom of the Will, in Journal of Philosophy, 1972, p.530.
8. P.527.
9. David L.Boyer, “Lucas, G”del and Astaire”, The Philosophical Quarterly, 1983, pp. 147-159.
10. Hao Wang, From Mathematics to Philosophy, London, 1974, p.315.
11. Douglas R.Hofstadter, G”del, Escher, Bach, New York, 1979, p.475.
12. Hao Wang, From Mathematics to Philosophy, London, 1974, pp.324-326.
13. Solomon Feferman, “Turing in the Land of O(z)”, in Rolf Herken ed., The Universal Turing Machine, Oxford, 1988, p.121.
14. A.M.Turing, “Systems of logic based on ordinals”, Proceedings of the London Mathematical Society, (2), 45, 1939, pp.161-228; reprinted in M.Davis, The Undecidable, New York, 1965; quoted by Solomon Feferman, op.cit., p.129.
15. I.J.Good, “G”del’s Theorem is a Red Herring”, British Journal for the Philosophy of Science, 19, 1968, pp. 357-358.
16. Judson C.Webb, Mechanism, Mentalism and Metamathematics; An Essay on Finitism, Dordrecht, 1980, p.230.
17. P.232, Webb’s italics.
18. Paul Benacerraf, “God, The Devil and G”del”, The Monist, 51, 1967, pp.
19. Hilary Putnam “Minds and Machines”, in Sidney Hook, ed., Dimensions of Mind: A Symposium, New York, 1960; reprinted in Kenneth M. Sayre and Frederick J.Crosson, eds., The Modeling of Mind, Notre Dame, 1963, pp. 255-271; and in A. R. Anderson, Minds and Machines, Prentice-Hall, 1964, pp. 43-59.
20. Hao Wang, From Mathematics to Philosophy, London, 1974, p.317.
21. Paul Benacerraf, “God, The Devil and G”del”, The Monist, 51, 1967, pp. 22-23; J.R. Lucas, “Satan Stultified”, The Monist, 52, 1967, pp. 152-3. 22. Hofstadter, 1979, p.578; cf. Charles S. Chihara, “On Alleged Refutations of Mechanism using G”del’s Incompleteness Results”, Journal of Philosophy, LXIX, no.17, 1972, p.526.
23. Anthony Hutton, “This G”del is Killing Me”, Philosophia, vol. 6, no.1, 1976, pp. 135-144.

 
—o0o—
Criticisms of the Gödelian Argument
J.J.C.Smart, “Gödel’s Theorem, Church’s Theorem, and Mechanism”, Synthese, 13, 1961.
J.J.C.Smart, “Man as a Physical Mechanism”, ch.VI of his Philosophy and Scientific Realism.
Hilary Putnam “Minds and Machines”, in Sidney Hook, ed., Dimensions of Mind. A Symposium, New York, 1960; reprinted in Kenneth M. Sayre and Frederick J. Crosson, eds., The Modeling of Mind, Notre Dame, 1963, pp. 255-271; and in A. R. Anderson, Minds and Machines, Prentice-Hall, 1964, pp. 43-59.
C.H. Whitely, “Minds, Machines and Gödel: a Reply to Mr. Lucas”, Philosophy, 37, 1962, pp.61-62.
Paul Benacerraf, “God, the Devil and Gödel”, The Monist, 1967, pp. 9-32.
I.J. Good, “Human and Machine Logic,” British Journal for the Philosophy of Science, 18, 1967, pp. 144-147.
I.J.Good, “Gödel’s Theorem is a Red Herring”, British Journal for the Philosophy of Science, 19, 1968, pp. 357-8.
David Lewis, “Lucas Against Mechanism”, Philosophy, XLIV, 1969, pp. 231-233.
David Coder, “Goedel’s Theorem and Mechanism”, Philosophy, XLIV, 1969, pp. 234-237, esp. p.236.
Jonathan Glover, Responsibility, London, 1970, p.31.
William Hanson, “Mechanism and Gödel’s Theorems,” British Journal for the Philosophy of Science, XXII, 1971.
D.C. Dennett, Review of The Freedom of the Will, Journal of Philosophy, 1972.
Charles S. Chihara, “On Alleged Refutations of Mechanism using Gödel’s Incompleteness Results”, Journal of Philosophy, LXIX, no.17, 1972.
Hao Wang, From Mathematics to Philosophy, London, 1974, pp.319, 320, 324-326.
A.J.P.Kenny in A.J.P.Kenny, H.C.Longuet-Higgins, J.R. Lucas and C.H.Waddington, The Nature of Mind, Edinburgh, 1976, p.75.
Anthony Hutton, “This Gödel is Killing Me”, Philosophia, vol. 6, no.1, 1976, pp. 135-144.
J.W. Thorp, “Free Will and Neurophysiological Determinism”, Oxford D.Phil. Thesis, 1976, p.79.
J.L. Mackie, Ethics: Inventing Right and Wrong, Penguin, 1977, p. 219.
David Lewis, “Lucas Against Mechanism II”, Canadian Journal of Philosophy, IX, 1979, pp. 373-376.
Douglas R.Hofstadter, Gödel, Escher, Bach, New York, 1979, p.475.
Emmanuel Q. Fernando, “Mathematical and Philosophical Implications of the Gödel Incompleteness Theorems”. M.A. Thesis, College of Arts and Sciences, University of the Philippines, Quezu City, September 1980.
Judson C. Webb, Mechanism, Mentalism and Metamathematics; An Essay on Finitism, Dordrecht, 1980, p.230.
G. Lee Bowie, “Lucas’ Number is Finally Up”, Journal of Philosophical Logic, 11, 1982, pp.279-285.
P.Sleazak, “Gödel’s Theorem and the Mind”, British Journal for the Philosophy of Science, XXXIII, 1982.
Rudy Rucker, “Gödel’s Theorem: The Paradox at the heart of modern man”, Popular Computing, February 1985, p.168.
David L. Boyer, “Lucas, Gödel and Astaire”, The Philosophical Quarterly, 1983, pp.147-159.
David Bostock, “Gödel and Determinism”, private communication, November, 1984.
Robert Kirk, “Mental Machinery and Gödel”, Synthese, 66, 1986, pp.437-452.
Other works are cited in The Freedom of the Will, pp. 174-6.

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  • Minds, Machines and Gödel — the original paper
  • Minds, Machines and Gödel
    First published in Philosophy, XXXVI, 1961, pp.(112)-(127); reprinted in The Modeling of Mind, Kenneth M.Sayre and Frederick J.Crosson, eds., Notre Dame Press, 1963, pp.[269]-[270]; and Minds and Machines, ed. Alan Ross Anderson, Prentice-Hall, 1954, pp.{43}-{59}.
    Gödel’s theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines. So also has it seemed to many other people: almost every mathematical logician I have put the matter to has confessed to similar thoughts, but has felt reluctant to commit himself definitely until he could see the whole argument set out, with all objections fully stated and properly met.1 This I attempt to do.
    Gödel’s theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot {44} be proved-in-the-system, but which we can see to be true. Essentially, we consider the formula which says, in effect, “This formula is unprovable-in-the-system”. If this formula were provable-in-the-system, we should have a contradiction: for if it were provablein-the-system, then it would not be unprovable-in-the-system, so that “This formula is unprovable-in-the-system” would be false: equally, if it were provable-in-the-system, then it would not be false, but would be true, since in any consistent system nothing false can be provedin-the-system, but only truths. So the formula “This formula is unprovable-in-the-system” is not provable-in-the-system, but unprovablein-the-system. Further, if the formula “This formula is unprovablein- the-system” is unprovable-in-the-system, then it is true that that [256] formula is unprovable-in-the-system, that is, “This formula is unprovable-in-the-system” is true.
    The foregoing argument is very fiddling, and difficult to grasp fully: it is helpful to put the argument the other way round, consider the possibility that “This formula is unprovable-in-the-system” might be false, show that that is impossible, and thus that the formula is true; whence it follows that it is unprovable. Even so, the argument remains persistently unconvincing: we feel that there must be a catch in it somewhere. The whole labour of Gödel’s theorem is to show that there is no catch anywhere, and that the result can (113) be established by the most rigorous deduction; it holds for all formal systems which are (i) consistent, (ii) adequate for simple arithmetic—i.e., contain the natural numbers and the operations of addition and multiplication—and it shows that they are incomplete— i.e., contain unprovable, though perfectly meaningful, formulae, some of which, moreover, we, standing outside the system, can see to be true.
    Gödel’s theorem must apply to cybernetical machines, because it is of the essence of being a machine, that it should be a concrete instantiation of a formal system. It follows that given any machine which is consistent and capable of doing simple arithmetic, there is a formula which it is incapable of producing as being true—i.e., the formula is unprovable-in-the-system-but which we can see to be true. It follows that no machine can be a complete or adequate model of the mind, that minds are essentially different from machines.
    We understand by a cybernetical machine an apparatus which performs a set of operations according to a definite set of rules. Normally we “programme” a machine: that is, we give it a set of instructions about what it is to do in each eventuality; and we feed in the initial “information” on which the machine is to perform its calculations. When we {45} consider the possibility that the mind might be a cybernetical mechanism we have such a model in view; we suppose that the brain is composed of complicated neural circuits, and that the information fed in by the senses is “processed” and acted upon or stored for future use. If it is such a mechanism, then given the way in which it is programmed—the way in which it is “wired up”—and the information which has been fed into it, the response—the “output”—is determined, and could, granted sufficient time, be calculated. Our idea of a machine is just this, that its behaviour is completely determined by the way it is made and the incoming “stimuli”: there is no possibility of its acting on its own: given a certain form of construction and a certain input of information, then it must act in a certain specific way. We, however, shall be concerned not with what a machine must do, but with what it can do. That is, instead [257] of considering the whole set of rules which together determine exactly what a machine will do in given circumstances, we shall consider only an outline of those rules, which will delimit the possible responses of the machine, but not completely. The complete rules will determine the operations completely at every stage; at every stage there will be a definite instruction, e.g., “If the number is prime and greater than two add one and divide by two: if it is not prime, divide by its smallest factor”: we, however, will consider the possibility of there being alternative instructions, e.g., “In a fraction you may divide top and bottom by any number which is a factor of both numerator and denominator”. In thus (114) relaxing the specification of our model, so that it is no longer completely determinist, though still entirely mechanistic, we shall be able to take into account a feature often proposed for mechanical models of the mind, namely that they should contain a randomizing device. One could build a machine where the choice between a number of alternatives was settled by, say, the number of radium atoms to have disintegrated in a given container in the past half- minute. It is prima facie plausible that our brains should be liable to random effects: a cosmic ray might well be enough to trigger off a neural impulse. But clearly in a machine a randomizing device could not be introduced to choose any alternative whatsoever: it can only be permitted to choose between a number of allowable alternatives. It is all right to add any number chosen at random to both sides of an equation, but not to add one number to one side and another to the other. It is all right to choose to prove one theorem of Euclid rather than another, or to use one method rather than another, but not to “prove” something which is not true, or to use a “method of proof” which is not valid. Any {46} randomizing devices must allow choices only between those operations which will not lead to inconsistency: which is exactly what the relaxed specification of our model specifies Indeed, one might put it this way: instead of considering what a completely determined machine must do, we shall consider what a machine might be able to do if it had a randomizing device that acted whenever there were two or more operations possible, none of which could lead to inconsistency.
    If such a machine were built to produce theorems about arithmetic (in many ways the simplest part of mathematics), it would have only a finite number of components, and so there would be only a finite number of types of operation it could do, and only a finite number of initial (115) assumptions it could operate on. Indeed, we can go further, and say that there would only be a definite number of types of operation, and of initial assumptions, that could be built into it. Machines are definite: anything which was indefinite or infinite we [258] should not count as a machine. Note that we say number of types of operation, not number of operations. Given sufficient time, and provided that it did not wear out, a machine could go on repeating an operation indefinitely: it is merely that there can be only a definite number of different sorts of operation it can perform.
    If there are only a definite number of types of operation and initial assumptions built into the system, we can represent them all by suitable symbols written down on paper. We can parallel the operation by rules (“rules of inference” or “axiom schemata”) allowing us to go from one or more formulae (or even from no formula at all) to another formula, and we can parallel the initial assumptions (if any) by a set of initial formulae (“primitive propositions”, “postulates” or “axioms”). Once we have represented these on paper, we can represent every single operation: all we need do is to give formulae representing the situation before and after the operation, and note which rule is being invoked. We can thus represent on paper any possible sequence of operations the machine might perform. However long, the machine went on operating, we could, give enough time, paper and patience, write down an analogue of the machine’s operations. This analogue would in fact be a formal proof: every operation of the machine is represented by the application of one of the rules: and the conditions which determine for the machine whether an operation can be performed in a certain situation, become, in our representation, conditions which settle whether a rule can be applied to a certain formula, i.e., formal conditions of applicability. Thus, construing our rules as rules of inference, we shall have a proof-sequence of {47} formulae, each one being written down in virtue of some formal rule of inference having been applied to some previous formula or formulae (except, of course, for the initial formulae, which are given because they represent initial assumptions built into the system). The conclusions it is possible for the machine to produce as being true will therefore correspond to the theorems that can be proved in the corresponding formal system. We now construct a Gödelian formula in this formal system. This formula cannot be proved-in-the- system. Therefore the machine cannot produce the corresponding formula as being true. But we can see that the Gödelian formula is true: any rational being could follow Gödel’s argument, and convince himself that the Gödelian formula, although unprovable-in-the-system, was nonetheless—-in fact, for that very reason—true. Now any mechanical model of the mind must include a mechanism which can enunciate truths of arithmetic, because this is something which minds can do: in fact, it is easy to produce mechanical models which will in many respects produce truths of arithmetic far [259] better than human beings can. But in this one respect they cannot do so well: in that for every machine there is a truth which it cannot produce as being true, but which a mind can. This shows that a machine cannot be a complete and adequate model of the mind. It cannot do everything that a mind can do, since however much it can do, there is always something which it cannot do, and a mind can. This is not to say that we cannot build a machine to simulate any desired piece of mind-like behaviour: it is only that we cannot build a machine to simulate every piece of mind-like behaviour. We can (or shall be able to one day) build machines capable of reproducing bits of mind-like behaviour, and indeed of outdoing the performances of human minds: but however good the machine is, and however much better (116) it can do in nearly all respects than a human mind can, it always has this one weakness, this one thing which it cannot do, whereas a mind can. The Gödelian formula is the Achilles’ heel of the cybernetical machine. And therefore we cannot hope ever to produce a machine that will be able to do all that a mind can do: we can never not even in principle, have a mechanical model of the mind.
    This conclusion will be highly suspect to some people. They will object first that we cannot have it both that a machine can simulate any piece of mind-like behaviour, and that it cannot simulate every piece. To some it is a contradiction: to them it is enough to point out that there is no contradiction between the fact that for any natural number there can be produced a greater number, and the fact that a number cannot {48} be produced greater than every number. We can use the same analogy also against those who, finding a formula their first machine cannot produce as being true, concede that that machine is indeed inadequate, but thereupon seek to construct a second, more adequate, machine, in which the formula can be produced as being true. This they can indeed do: but then the second machine will have a Gödelian formula all of its own, constructed by applying Gödel’s procedure to the formal system which represents its (the second machine’s) own, enlarged, scheme of operations. And this formula the second machine will not be able to produce as being true, while a mind will be able to see that it is true. And if now a third machine is constructed, able to do what the second machine was unable to do, exactly the same will happen: there will be yet a third formula, the Gödelian formula for the formal system corresponding to the third machine’s scheme of operations, which the third machine is unable to produce as being true, while a mind will still be able to see that it is true. And so it will go on. However complicated a machine we construct, it will, if it is a machine, correspond to a formal system, which in turn will be liable to the Gödel procedure [260] for finding a formula unprovable-in-that- system. This formula the machine will be unable to produce as being true, although a mind can see that it is true. And so the machine will still not be an adequate model of the mind. We are trying to produce a model of the mind which is mechanical—which is essentially “dead”—but the mind, being in fact “alive”, can always go one better than any formal, ossified, dead, system can. Thanks to Gödel’s theorem, the mind always has the last word.
    A second objection will now be made. The procedure whereby the Gödelian formula is constructed is a standard procedure—only so could we be sure that a Gödelian formula can be constructed for every formal system. But if it is a standard procedure, then a machine should be able to be programmed to carry it out too. We could construct a machine with the usual operations, and in addition an (117) operation of going through the Gödel procedure, and then producing the conclusion of that procedure as being true; and then repeating the procedure, and so on, as often as required. This would correspond to having a system with an additional rule of inference which allowed one to add, as a theorem, the Gödelian formula of the rest of the formal system, and then the Gödelian formula of this new, strengthened formal system, and so on. It would be tantamount to adding. to the original formal system an infinite sequence of axioms, each the Gödelian formula of the system hitherto obtained. Yet even so, the matter is not settled: for the machine with a Gödelizing {49} operator, as we might call it, is a different machine from the machines without such an operator; and, although the machine with the operator would be able to do those things in which the machines without the operator were outclassed by a mind, yet we might expect a mind, faced with a machine that possessed a Gödelizing operator, to take this into account, and out-Gödel the new machine, Gödelizing operator and all. This has, in fact, proved to be the case. Even if we adjoin to a formal system the infinite set of axioms consisting of the successive Gödelian formulae, the resulting system is still incomplete, and contains a formula which cannot be proved-in-the-system, although a rational being can, standing outside the system, see that it is true.2 We had expected this, for even if an infinite set of axioms were added, they would have to be specified by some finite rule or specification, and this further rule or specification could then be taken into account by a mind considering the enlarged formal system. In a sense, just because the mind has the last word, it can always pick a hole in any formal system presented to it as a model of its own workings. The [261] mechanical model must be, in some sense, finite and definite: and then the mind can always go one better.
    This is the answer to one objection put forward by Turing.3 He argues that the limitation to the powers of a machine do not amount to anything much. Although each individual machine is incapable of getting the right answer to some questions, after all each individual human being is fallible also: and in any case “our superiority can only be felt on such an occasion in relation to the one machine over which we have scored our petty triumph. There would be no question of triumphing simultaneously over all machines.” But this is not the point. We are not discussing whether machines or minds are superior, but whether they are the same. In some respect machines are undoubtedly superior to human minds; and the question on which they are stumped is admittedly, a rather niggling, even (118) trivial, question. But it is enough, enough to show that the machine is not the same as a mind. True, the machine can do many things that a human mind cannot do: but if there is of necessity something that the machine cannot do, though the mind can, then, however trivial the matter is, we cannot equate the two, and cannot hope ever to have a mechanical model that will adequately represent the mind. Nor does it signify that it is only an individual machine we have triumphed over: for the triumph is not over only an individual machine, but over any individual that anybody cares to specify—in Latin {50} quivis or quilibet, not quidam—and a mechanical model of a mind must be an individual machine. Although it is true that any particular “triumph” of a mind over a machine could be “trumped” by another machine able to produce the answer the first machine could not produce, so that “there is no question of triumphing simultaneously over all machines”, yet this is irrelevant. What is at issue is not the unequal contest between one mind and all machines, but whether there could be any, single, machine that could do all a mind can do. For the mechanist thesis to hold water, it must be possible, in principle, to produce a model, a single model, which can do everything the mind can do. It is like a game.4 The mechanist has first turn. He produces a—any, but only a definite one—mechanical model of the mind. I point to something that it cannot do, but the mind can. The mechanist is free to modify his example, but each time he does so, I am entitled to look for defects in the revised model. If the mechanist can devise a model that I cannot find fault with, his [262] thesis is established: if he cannot, then it is not proven: and since—as it turns out-he necessarily cannot, it is refuted. To succeed, he must be able to produce some definite mechanical model of the mind—anyone he likes, but one he can specify, and will stick to. But since he cannot, in principle cannot, produce any mechanical model that is adequate, even though the point of failure is a minor one, he is bound to fail, and mechanism must be false.
    Deeper objections can still be made. Gödel’s theorem applies to deductive systems, and human beings are not confined to making only deductive inferences. Gödel’s theorem applies only to consistent systems, and one may have doubts about how far it is permissible to assume that human beings are consistent. Gödel’s theorem applies only to formal systems, and there is no a priori bound to human ingenuity which rules out the possibility of our contriving some replica of humanity which was not representable by a formal system.
    Human beings are not confined to making deductive inferences, and it has been urged by C.G. Hempel5 and Hartley Rogers6 that a fair model of the mind would have to allow for the possibility of making non-deductive inferences, and these might provide a way of escaping the Gödel result. Hartley Rogers makes the specific suggestion that the {51} machine should be programmed to entertain various propositions which had not been proved or disproved, and on occasion to add them to its list of axioms. Fermat’s last theorem or Goldbach’s conjecture might thus be added. If subsequently their inclusion was found to lead to a contradiction, they would be dropped again, and indeed in those circumstances their negations would be added to the list of theorems. In this sort of way a machine might well be constructed which was able to produce as true certain formulae which could not be proved from its axioms according to its rules of inference. And therefore the method of demonstrating the mind’s superiority over the machine might no longer work.
    The construction of such a machine, however, presents difficulties. It cannot accept all unprovable formulae, and add them to its axioms, or it will find itself accepting both the Gödelian formula and its negation, and so be inconsistent. Nor would it do if it accepted the first of each pair of undecidable formulae, and, having added that to its axioms, would no longer regard its negation as undecidable, and so would never accept it too: for it might happen on the wrong member of the pair: it might accept the negation of the Gödelian formula rather than the Gödelian formula itself. And the system constituted [263] by a normal set of axioms with the negation of the Gödelian formula adjoined, although not inconsistent, is an unsound system, not admitting of the natural interpretation. It is something like non- Desarguian geometries in two dimensions: not actually inconsistent, but rather wrong, sufficiently much so to disqualify it from serious consideration. A machine which was liable to infelicities of that kind would be no model for the human mind.
    It becomes clear that rather careful criteria of selection of unprovable formulae will be needed. Hartley Rogers suggests some possible ones. But once we have rules generating new axioms, even if the axioms generated are only provisionally accepted, and are liable to be dropped again if they are found to lead to inconsistency, then we can set about doing a Gödel on this system, as on any other. We are in the same case as when we had a rule generating the infinite set of Gödelian formulae as axioms. In short, however a machine is designed, it must proceed either at random or according to definite rules. In so far as its procedure is random, we cannot outsmart it: (120) but its performance is not going to be a convincing parody of intelligent behaviour: in so far as its procedure is in accordance with definite rules, the Gödel method can {52} be used to produce a formula which the machine, according to those rules, cannot assert as true, although we, standing outside the system, can see it to be true.7
    Gödel’s theorem applies only to consistent systems. All that we can prove formally is that if the system is consistent, then the Gödelian formula is unprovable-in-the-system. To be able to say categorically that the Gödelian formula is unprovable-in- the-system, and therefore true, we must not only be dealing with a consistent system, but be able to say that it is consistent. And, as Gödel showed in his second theorem—a corollary of his first—it is impossible to prove in a consistent system that that system is consistent. Thus in order to fault the machine by producing a formula of which we can say both that it is true and that the machine cannot produce it as true, we have to be able to say that the machine (or, rather, its corresponding formal system) is consistent; and there is no absolute proof of this. All we can do is to examine the machine and see if it appears consistent. There always remains the possibility of some inconsistency not yet detected. At best we can say that the machine is consistent, provided we are. But by what right can we do this? Gödel’s second [264] theorem seems to show that a man cannot assert his own consistency, and so Hartley Rogers8 argues that we cannot really use Gödel’s first theorem to counter the mechanist thesis unless we can say that “there are distinctive attributes which enable a human being to transcend this last limitation and assert his own consistency while still remaining consistent”.
    A man’s untutored reaction if his consistency is questioned is to affirm it vehemently: but this, in view of Gödel’s second theorem, is taken by some philosophers as evidence of his actual inconsistency. Professor Putnam9 has suggested that human beings are machines, but inconsistent machines. If a machine were wired to correspond to an inconsistent system, then there would be no well-formed formula which it could not produce as true; and so in no way could it be proved to be inferior to a human being. Nor could we make its inconsistency a reproach to it—are not men inconsistent too? Certainly women are, and politicians; and {53} even male non-politicians (121) contradict themselves sometimes, and a single inconsistency is enough to make a system inconsistent.
    The fact that we are all sometimes inconsistent cannot be gainsaid, but from this it does not follow that we are tantamount to inconsistent systems. Our inconsistencies are mistakes rather than set policies. They correspond to the occasional malfunctioning of a machine, not its normal scheme of operations. Witness to this that we eschew inconsistencies when we recognize them for what they are. If we really were inconsistent machines, we should remain content with our inconsistencies, and would happily affirm both halves of a contradiction. Moreover, we would be prepared to say absolutely anything—which we are not. It is easily shown10 that in an inconsistent formal system everything is provable, and the requirement of consistency turns out to be just that not everything can be proved in it—it is not the case that “anything goes.” This surely is a characteristic of the mental operations of human beings: they are selective: they do discriminate between favoured—true—and unfavoured— false—statements: when a person is prepared to say anything, and is prepared to contradict himself without any qualm or repugnance, then he is adjudged to have “lost his mind”. Human beings, although not perfectly consistent, are not so much inconsistent as fallible.
    A fallible but self-correcting machine would still be subject to Gödel’s results. Only a fundamentally inconsistent machine would [265] escape. Could we have a fundamentally inconsistent, but at the same time self- correcting machine, which both would be free of Gödel’s results and yet would not be trivial and entirely unlike a human being? A machine with a rather recherché: inconsistency wired into it, so that for all normal purposes it was consistent, but when presented with the Gödelian sentence was able to prove it?
    There are all sorts of ways in which undesirable proofs might be obviated. We might have a rule that whenever we have proved p and not-p, we examine their proofs and reject the longer. Or we might arrange the axioms and rules of inference in a certain order, and when a proof leading to an inconsistency is proffered, see what axioms and rules are required for it, and reject that axiom or rule which comes last in the ordering. In some such way as this we could have an inconsistent system, with a stop-rule, so that the inconsistency was never allowed to come out in the form of an inconsistent formula.
    The suggestion at first sight seems attractive: yet there is something deeply wrong. Even though we might preserve the facade of consistency {54} by having a rule that whenever two inconsistent formulae (122) appear we were to reject the one with the longer proof, yet such a rule would be repugnant in our logical sense. Even the less arbitrary suggestions are too arbitrary. No longer does the system operate with certain definite rules of inference on certain definite formulae. Instead, the rules apply, the axioms are true, provided . . . we do not happen to find it inconvenient. We no longer know where we stand. One application of the rule of Modus Ponens may be accepted while another is rejected: on one occasion an axiom may be true, or another apparently false. The system will have ceased to be a formal logical system, and the machine will barely qualify for the title of a model for the mind. For it will be far from resembling the mind in its operations: the mind does indeed try out dubious axioms and rules of inference; but if they are found to lead to contradiction, they are rejected altogether. We try out axioms and rules of inference provisionally—true: but we do not keep them, once they are found to lead to contradictions. We may seek to replace them with others, we may feel that our formalization is at fault, and that though some axiom or rule of inference of this sort is required, we have not been able to formulate it quite correctly: but we do not retain the, faulty formulations without modification, merely with the proviso that when the argument leads to a contradiction we refuse to follow it. To do this would be utterly irrational. We should be in the position that on some occasions when supplied with the premisses of a Modus Ponens, say, we applied the rule and allowed the conclusion, and [266] on other occasions we refused to apply the rule, and disallowed the conclusion. A person, or a machine, which did this without being able to give a good reason for so doing, would be accounted arbitrary and irrational. It is part of the concept of “arguments” or “reasons” that they are in some sense general and universal: that if Modus Ponens is a valid method of arguing when I am establishing a desired conclusion, it is a valid method also when you, my opponent, are establishing a conclusion I do not want to accept. We cannot pick and choose the times when a form of argument is to be valid; not if we are to be reasonable. It is of course true, that with our informal arguments, which are not fully formalized, we do distinguish between arguments which are at first sight similar, adding further reasons why they are nonetheless not really similar: and it might be maintained that a {55} machine might likewise be entitled to distinguish between arguments at first sight similar, if it had good reason for doing so. And it might further be maintained that the machine had good reason for rejecting those patterns of argument it did reject, indeed the best of reasons, namely the avoidance of contradiction. But that, if it is a reason at all, is too good a reason. We do not lay it to a man’s credit that he avoids contradiction merely by refusing to accept those arguments which would lead him to it, for no other (123) reason than that otherwise he would be led to it. Special pleading rather than sound argument is the name for that type of reasoning. No credit accrues to a man who, clever enough to see a few moves of argument ahead, avoids being brought to acknowledge his own inconsistency, by stonewalling as soon as he sees where the argument will end. Rather, we account him inconsistent too, not, in his case, because he affirmed and denied the same proposition, but because he used and refused to use the same rule of inference. A stop-rule on actually enunciating an inconsistency is not enough to save an inconsistent machine from being called inconsistent.
    The possibility yet remains that we are inconsistent, and there is no stop-rule, but the inconsistency is so recherché: that it has never turned up. After all, naive set-theory, which was deeply embedded in common- sense ways of thinking did turn out to be inconsistent. Can we be sure that a similar fate is not in store for simple arithmetic too? In a sense we cannot, in spite of our great feeling of certitude that our system of whole numbers which can be added and multiplied together is never going to prove inconsistent. It is just conceivable we might find we had formalized it incorrectly. If we had, we should try and formulate anew our intuitive concept of number, as we have our intuitive concept of a set. If we did this, we should of course recast our system: our present axioms and rules of inference would [267] be utterly rejected: there would be no question of our using and not using them in an “inconsistent” fashion. We should, once we had recast the system, be in the same position as we are now, possessed of a system believed to be consistent, but not provably so. But then could there not be some other inconsistency? It is indeed a possibility. But again no inconsistency once detected will be tolerated. We are determined not to be inconsistent, and are resolved to root out inconsistency, should any appear. Thus, although we can never be completely certain or completely free of the risk of having to think out our mathematics again, the ultimate position must be one of two: either we have a system of simple arithmetic which to the best of our knowledge and belief is consistent: or there is no such system possible. In the former case we are in the same position as at present: in the {56} latter, if we find that no system containing simple arithmetic can be free of contradictions, we shall have to abandon not merely the whole of mathematics and the mathematical sciences, but the whole of thought.
    It may still be maintained that although a man must in this sense assume, he cannot properly affirm, his own consistency without thereby belying his words. We may be consistent; indeed we have every reason to hope that we are: but a necessary modesty forbids us from saying so. Yet this is not quite what Gödel’s second theorem states. Gödel has shown that in a consistent system a formula (124) stating the consistency of the system cannot be proved in that system. It follows that a machine, if consistent, cannot produce as true an assertion of its own consistency: hence also that a mind, if it were really a machine, could not reach the conclusion that it was a consistent one. For a mind which is not a machine no such conclusion follows. All that Gödel has proved is that a mind cannot produce a formal proof of the consistency of a formal system inside the system itself: but there is no objection to going outside the system and no objection to producing informal arguments for the consistency either of a formal system or of something less formal and less systematized. Such informal arguments will not be able to be completely formalized: but then the whole tenor of Gödel’s results is that we ought not to ask, and cannot obtain, complete formalization. And although it would have been nice if we could have obtained them, since completely formalized arguments are more coercive than informal ones, yet since we cannot have all our arguments cast into that form, we must not hold it against informal arguments that they are informal or regard them all as utterly worthless. It therefore seems to me both proper and reasonable for a mind to assert its own consistency: proper, because although machines, as we might have expected, are [268] unable to reflect fully on their own performance and powers, yet to be able to be self-conscious in this way is just what we expect of minds: and reasonable, for the reasons given. Not only can we fairly say simply that we know we are consistent, apart from our mistakes, but we must in any case assume that we are, if thought is to be possible at all; moreover we are selective, we will not, as inconsistent machines would, say anything and everything whatsoever: and finally we can, in a sense, decide to be consistent, in the sense that we can resolve not to tolerate inconsistencies in our thinking and speaking, and to eliminate them, if ever they should appear, by withdrawing and cancelling one limb of the contradiction.
    We can see how we might almost have expected Gödel’s theorem to distinguish self-conscious beings from inanimate objects. The essence of {57} the Gödelian formula is that it is self-referring. It says that “This formula is unprovable-in-this-system”. When carried over to a machine, the formula is specified in terms which depend on the particular machine in question. The machine is being asked a question about its own processes. We are asking it to be self-conscious, and say what things it can and cannot do. Such questions notoriously lead to paradox. At one’s first and simplest attempts to philosophize, one becomes entangled in questions of whether when one knows something one knows that one knows it, and what, when one is thinking of oneself, is being thought about, and what is doing the thinking. After one has been puzzled and bruised by this (125) problem for a long time, one learns not to press these questions: the concept of a conscious being is, implicitly, realized to be different from that of an unconscious object. In saying that a conscious being knows something, we are saying not only that he knows it, but that he knows that he knows it, and that he knows that he knows that he knows it, and so on, as long as we care to pose the question: there is, we recognize, an infinity here, but it is not an infinite regress in the bad sense, for it is the questions that peter out, as being pointless, rather than the answers. The questions are felt to be pointless because the concept contains within itself the idea of being able to go on answering such questions indefinitely. Although conscious beings have the power of going on, we do not wish to exhibit this simply as a succession of tasks they are able to perform, nor do we see the mind as an infinite sequence of selves and super-selves and super-superselves. Rather, we insist that a conscious being is a unity, and though we talk about parts of the mind, we do so only as a metaphor, and will not allow it to be taken literally.
    The paradoxes of consciousness arise because a conscious being can be aware of itself, as well as of other things, and yet cannot [269] really be construed as being divisible into parts. It means that a conscious being can deal with Gödelian questions in a way in which a machine cannot, because a conscious being can both consider itself and its performance and yet not be other than that which did the performance. A machine can be made in a manner of speaking to “consider” its own performance, but it cannot take this “into account” without thereby becoming a different machine, namely the old machine with a “new part” added. But it is inherent in our idea of a conscious mind that it can reflect upon itself and criticize its own performances, and no extra part is required to do this: it is already complete, and has no Achilles’ heel.
    The thesis thus begins to become more a matter of conceptual analysis {58}than mathematical discovery. This is borne out by considering another argument put forward by Turing.11 So far, we have constructed only fairly simple and predictable artefacts. When we increase the complexity of our machines there may, perhaps, be surprises in store for us. He draws a parallel with a fission pile. Below a certain “critical” size, nothing much happens: but above the critical size, the sparks begin to fly. So too, perhaps, with brains and machines. Most brains and all machines are, at present, “subcritical”—they react to incoming stimuli in a stodgy and uninteresting way, have no ideas of their own, can produce only stock responses —but a few brains at present, and possibly some machines in the future, are super-critical, and scintillate on their own account. (126) Turing is suggesting that it is only a matter of complexity, and that above a certain level of complexity a qualitative difference appears, so that 44 super-critical” machines will be quite unlike the simple ones hitherto envisaged.
    This may be so. Complexity often does introduce qualitative differences. Although it sounds implausible, it might turn out that above a certain level of complexity, a machine ceased to be predictable, even in principle, and started doing things on its own account, or, to use a very revealing phrase, it might begin to have a mind of its own. It might begin to have a mind of its own. It would begin to have a mind of its own when it was no longer entirely predictable and entirely docile, but was capable of doing things which we recognized as intelligent, and not just mistakes or random shots, but which we had not programmed into it. But then it would cease to be a machine, within the meaning of the act. What is at stake in the mechanist debate is not how minds are, or might be, brought into being, but how they operate. It is essential for the mechanist thesis that the mechanical model of the mind shall operate according [270] to “mechanical principles”, that is, that we can understand the operation of the whole in terms of the operations of its parts, and the operation of each part either shall be determined by its initial state and the construction of the machine, or shall be a random choice between a determinate number of determinate operations. If the mechanist produces a machine which is so complicated that this ceases to hold good of it, then it is no longer a machine for the purposes of our discussion, no matter how it was constructed. We should say, rather, that he had created a mind, in the same sort of sense as we procreate people at present. There would then be two ways of bringing new minds into the world, the traditional way, by begetting children born of women, and a new way by constructing very, very complicated systems of, say, valves {59} and relays. When talking of the second way, we should take care to stress that although what was created looked like a machine, it was not one really, because it was not just the total of its parts. One could not tell what it was going to do merely by knowing the way in which it was built up and the initial state of its parts: one could not even tell the limits of what it could do, for even when presented with a Gödel-type question, it got the answer right. In fact we should say briefly that any system which was not floored by the Gödel question was eo ipso not a Turing machine, i.e., not a machine within the meaning of the act.
    If the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy. Since the time of Newton, the bogey of mechanist determinism has obsessed philosophers. If we were to be scientific, it seemed that we must look on human beings as (127) determined automata, and not as autonomous moral agents; if we were to be moral, it seemed that we must deny science its due, set an arbitrary limit to its progress in understanding human neurophysiology, and take refuge in obscurantist mysticism. Not even Kant could resolve the tension between the two standpoints. But now, though many arguments against human freedom still remain, the argument from mechanism, perhaps the most compelling argument of them all, has lost its power. No longer on this count will it be incumbent on the natural philosopher to deny freedom in the name of science: no longer will the moralist feel the urge to abolish knowledge to make room for faith. We can even begin to see how there could be room for morality, without its being necessary to abolish or even to circumscribe the province of science. Our argument has set no limits to scientific enquiry: it will still be possible to investigate the working of the brain. It will still be possible to produce mechanical models of the mind. Only, now we can see that no mechanical model will be completely adequate, nor any explanations [271] in purely mechanist terms. We can produce models and explanations, and they will be illuminating: but, however far they go, there will always remain more to be said. There is no arbitrary bound to scientific enquiry: but no scientific enquiry can ever exhaust the infinite variety of the human mind.12
    1. See A. M. Turing, “Computing Machinery and Intelligence,” Mind, 1950, pp. 433-60, reprinted in The World of Mathematics, edited by James R. Newmann, pp. 2099-2123; and K. R. Popper, “Indeterminism in Quantum Physics and Classical Physics,” British Journal for Philosophy of Science, 1 (1951), 179-88. The question is touched upon by Paul Rosenbloom; Elements of Mathematical Logic, pp. 207-8; Ernest Nagel and James R. Newmann, Gödel’s Proof, pp. 100-2; and by Hartley Rogers, Theory of Recursive Functions and Effective Computability (mimeographed), 1957, Vol. 1, pp. 152 ff. 2. Gödel’s original proof applies; v. I init. and 6 init. of his Lectures at the Institute of Advanced Study, Princeton, N.J., U.S.A., 1934. 3. Mind, 1950, pp. 444-5; Newman, p. 2110. 4. For a similar type of argument, see J. R. Lucas: “The Lesbian Rule”; Philosophy, (July 1955) pp. 202-206; and “On Not Worshipping Facts”; The Philosophical Quarterly, April 1958, p. 144. 5. In private conversation. 6. Theory of Recursive Functions and Effective Computability, 1957, Vol. 1, pp. 152 ff. 7. Gödel’s original proof applies if the rule is such as to generate a primitive recursive class of additional formulae; v. I init. and 6 init. of his Lectures at the Institute of Advanced Study, Princeton, N.J., U.S.A., 1934. It is in fact sufficient that the class be recursively enumerable. See Barkley Rosser: “Extensions of some theorems of Gödel and Church,” Journal of Symbolic Logic, 1, 1936, pp. 87-91. 8. Op. cit., p. 154. 9. University of Prineeton, N.J., U.S.A. in private conversation. 10. See, e.g., Alonzo Church: Introduction to Mathematical Logic, Princeton, Vol.1, 17, p. 108. 11. Mind, 1950, p. 454; Newman, pp. 2117-18. 12.
    Some objections by Benacerraf and Putman are considered in “Satan Stultified”
    A fuller account, in which further objections are considered, is given in J.R.Lucas, The Freedom of the Will, Oxford, 1970. Most critics concentrate their fire on “Minds, Machines and Gödel”, without looking at The Freedom of the Will. In recent years it has been out of print. But under a new intitative by OUP, it is now available again. Single copies are printed on a one-off basis. I commend it to those who think there are holes in the article reprinted here. (In the fulness of time I hope to scan the relevant pages and make them available on the Web: but I have many other pressing calls on my time.)
    See also Turn Over the Page a talk I gave on 25/5/96 at a BSPS conference in Oxford

    and “Minds, Machines and Gödel: A Retrospect”, in P.J.R.Millican and A.Clark, eds., Machines and Thought: The Legacy of Alan Turing, Oxford, 1996, pp.103-124.
    A full discussion of the issues raised is now available in Etica e Politica, 2003.
    GoogleScholar gives a large number (ca. 242) of references to critical discussions.
    an old list of criticisms and discussions of the Gödelian argument. Click here to return to home page
    Click here to return to bibliography

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