On 02 May 2016, at 03:38, Maxine Sheets-Johnstone wrote:

To all concerned colleagues,I appreciate the fact that discussions should be conversations aboutissues,but this particular issue and in particular the critique cited in mypostingwarrant extended exposition in order to show the reasoning upholdingthe critique.I am thus quoting from specific articles, the firstphenomenological, the secondanalytic-logical--though they are obviously complementary as befitsdiscussionsin phenomenology and the life sciences. EXCERPT FROM: SELF-REFERENCE AND GÖDEL'S THEOREM: A HUSSERLIAN ANALYSIS Husserl Studies 19 (2003), pages 131-151. Albert A. JohnstoneThe aim of this article is to show that a Husserlian approach to theLiar paradoxes and to their closely related kin discloses theillusory nature of these difficulties. Phenomenological meaninganalysis finds the ultimate source of mischief to be circulardefinition, implicit or explicit. Definitional circularity lies atthe root both of the self-reference integral to the statements thatgenerate Liar paradoxes, and of the particular instances ofpredicate criteria featured in the Grelling paradox as well as inthe self-evaluating Gödel sentence crucial to Gödel's theorem.Since the statements thereby generated turn out on closer scrutinyto be vacuous and semantically nonsensical, their rejection fromreasonable discourse is both warranted and imperative. Naturallyenough, their exclusion dissolves the various problems created bytheir presence. . . .VII: THE GOEDEL SENTENCEFollowing a procedure invented by Gödel, one may assign numbers insome orderly way as names or class-numbers to each of the variousclasses of numbers (the prime numbers, the odd numbers, and so on).Some of these class-numbers will qualify for membership in the classthey name; others will not. For instance, if the number 41 shouldhappen to be the class-number that names the class of numbers thatare divisible by 7, then since 41 does not have the property ofbeing divisible by 7, the class-number 41 would not be a member ofthe class it names.Now, consider the class-number of the class of class-numbers thatare members of the class they name. Does it have the definingproperty of the class it names? The question is unanswerable. Sincethe defining property of the class is that of being a class-numberthat is a member of the class it names, the necessary and sufficientcondition for the class-number in question to be a member of theclass it names turns out to be that it be a member of the class itnames. In short, the number is a member if and only if it is amember. The criterion is circular--defined in terms of what was tobe defined--and consequently not a criterion at all since itprovides no way of determining whether or not the number is a member.The situation is obviously similar for the class-number of thecomplementary class of class-numbers--those that do not have thedefining property of the class they name--since the criteria in thetwo cases are logically interdependent. The criterion of membershipis likewise defined in circular fashion, and hence is vacuous. Inaddition, the criterion postulates an absurd analytic equivalence,that of the defining property with its negative. The question ofwhether the class-number is a member of the class it names isunanswerable, with the result that any proposed answer is neithertrue nor false. In addition, of course, any answer would generateparadox: the number has the requisite defining property if and onlyif it does not have it.As might be expected, the situation is not significantly differentfor the class-number of classes of which the definition involvessemantic predicates. Consider, for instance, the class of class-numbers of which it is provable that they are members of the classthey name. The question of whether the class-number of the class isa member of the class it numbers is undecidable. The possession bythe class-number of the property requisite for membership isconditional upon the question of whether it provably possesses theproperty, with the result that the question can have no answer.Otherwise stated, the number has the defining property of the classit names if and only if it provably has that property. In thesecircumstances, the explanation of what it means for the class-numberto have the property has to be circular in that it must definehaving the property in terms of having the property. The vacuitythat results is hidden somewhat by the presence of the requirementof provability, but while provability might count as a necessarycondition, in the present case it cannot be a sufficient one. Infact, its presence creates a semantically absurd situation: theanalytic equivalence of having the property and provably having it.The statement of the possession of the property by the class-numberin question is consequently both vacuous and semantically absurd,hence an undecidable pseudo-statement.The analytic equivalence of the number's having the property andprovably having it has a further and quite interesting consequence.In principle, since the equivalence is analytic, it explains what itmeans to say that the class-number in question has the requisiteproperty, that is, it explains what is being said by the statementthat attributes the property to the number. What the statement issaying, according to the equivalence, is that it is provable thatthe number has the property, which is to say, it is saying of itselfthat it is provable. Thus, the statement is self-evaluating. It isnot, strictly speaking, self-referential since it contains nodesignator, and so cannot refer to itself. However, it mirrors theself-referential statements of the sort discussed earlier in that itpredicates a semantic property of itself (or at least purports to doso).In these circumstances, it is not overly surprising to find that asentence having a vacuously defined semantic predicate ofprovability is ambiguous or leads a double life. It may be used toexpress either of two statements, a pseudo-statement that purportsto evaluate itself as provable, or, a genuine statement thatevaluates the pseudo-statement, which genuine statement is, ofcourse, false since a pseudo-statement is in principle not provable.The two statements, genuine and pseudo, are not the same statement.The two have distinct truth-values,

Why?

but the basic point is that they differ in intended meaning. In thepseudo-statement, the statement itself (that a particular number hasa particular property) is a part of the meaning of the pseudo-statement, while in the genuine (but false) statement, it is not.An analogous situation obtains in the case of other classesinvolving semantic predicates. If the term 'heterological' thatfigures in the Grelling Paradox were defined as applying to thosewords of which it is false that they are heterological, then theresulting Grelling statement (the statement that 'heterological' isheterological) could be plausibly interpreted to be self-evaluating.It would be analytically equivalent to the statement that it isfalse that 'heterological' is heterological--an equivalence that maybe read as saying that the Grelling statement says of itself that itis false. This second statement would, of course, find itselfexpressed by a sentence that leads a double life.Of particular interest for the purpose of understanding the errorthat invalidates Gödel's theorem is the case of the class-numberthat names the class of class-numbers that are not provably membersof the class they name. Once again, the question as to whether theclass-number that names this class is a member of the class it namesis unanswerable. The statement that the class-number possesses therequired defining characteristic is a criterially deficientpredication, and hence a pseudo-statement. In addition, thestatement is analytically equivalent to the statement that the class-number's possession of the defining characteristic is not provable,and so may be viewed as saying of itself that it is not provable. Itis thus self-evaluating, and when stated in this form, it isexpressed by a sentence that leads a double life. As a result, anyformal system that admits and purports to accommodate a criteriallydeficient predication of the sort will also require the elaboratesupplementary machinery found necessary to accommodate self-referential statements: a three-valued logic, a procedure fordetermining which instantiations of predicates (or substitutionsinto propositional functions) produce pseudo-statements, and somenotational device for distinguishing pseudo-statements from thegenuine statements that are their sentential doubles. As we shallnow see, in view of the similarity in structure of the abovestatement to the Gödel sentence, analogous remarks apply to thelatter.VII. THE GÖDEL SENTENCEIn his well-known theorem Kurt Gödel purports to show that anyformal system of classical logic equivalent to that of PrincipiaMathematica to which arithmetic constants and the axioms ofarithmetic (Peano's) have been added, will contain sentences thatare undecidable--that is, sentences such that neither they nor theirnegations are provable within the system. To this end he introducesa provability predicate defined syntactically as membership in theset of sentences that are immediate consequences of the axiom-sentences. Since the provability predicate applies to sentencesrather than statements, to avoid confusion it is best termed 'aderivability predicate'. As in the arguments of the previoussection, Gödel has a number assigned as a name to each class ofnumbers according to its rank in an ordering of the various classesof numbers. Roughly characterized, the undecidable sentence figuringin the theorem (the Gödel sentence) states that a particular class-number satisfies a particular one-place propositional function thatdefines a class of numbers. A little more precisely, it states thata particular class-number has the defining characteristic of theclass it numbers, which class is the class of class-numbers suchthat the sentences stating that the class-numbers possess thedefining characteristics of the classes they name are notderivable. In his informal introduction to his theorem, Gödelpoints out that the sentence may be read as stating via its Gödelnumber that a particular sentence, itself, is not derivable.The crucial line of reasoning in the theorem strongly resembles theone found in the Liar. It runs roughly as follows: if the sentencewere derivable, it would have to be true, hence say something true,and hence, as it says, not be derivable--which contradicts theassumption of its derivability; if the negation of the sentence werederivable, then since the sentence states its underivability, itwould have to be not underivable, hence derivable--with the resultthat both the sentence and its negation would be derivable, acontradiction. As with the Liar, each of two possible alternativesgenerates a contradiction, although in the present case theconsequence is not paradox but undecidability-- undecidability inthe form of a sentence of which neither its truth nor its falsity isderivable in the system. Gödel reasons that since the undecidablesentence apparently states something true, its own underivability,the system contains underivable true sentences, and hence isincomplete.The Gödel sentence is concerned with derivability rather thanprovability, or sentences rather than statements. As a result onemay plausibly question whether it is vulnerable to the criticismsdirected above against criterially circular predications and self-evaluations. While the Gödel sentence clearly differs from thelatter, it is possible nevertheless to raise the question of itslegitimacy. Gödel himself simply assumes that the sentence islegitimate--which, of course, it is in the narrow sense that itconforms to the formation rules of the system in which it figures.However, it does not follow that it is legitimate in the broadersense that the interpreted sentence makes sense. As we saw earlierwith self-referential statements and criterially circularpredications, sentences that are apparently well-formed may in factexpress nonsense. The Gödel sentence may well express just such apseudo-statement, and have nevertheless been admitted into theformal system through an inadequacy of the formation rules. Gödeldismisses the possibility of faulty circularity on the grounds thatthe sentence states only that a certain well-defined formula isunprovable, which formula turns out after the fact to be the onethat expresses the proposition itself. Yet, an answer of the sortwill not do. Where circularity results from a substitution, beingadventitious and well-formed according to the rules do nothing toremove the circularity. A statement with a circularly definedpredicate is semantically vacuous, and hence not a genuinestatement. Thus, the question of the meaning of the sentence, thestatement it expresses, calls for serious examination.A first rather curious fact that more careful scrutiny brings tolight is that the most obvious reasons for thinking the sentencemeaningful are actually inconclusive. For instance, it might befound tempting to argue as follows: that any particular string ofsymbols is either derivable from the axiom-strings or not, and hencesince the Gödel sentence asserts that a particular string is notderivable, whether true or not, it must at least be meaningful.However, the reasoning begs the point at issue. If the Gödelsentence is not meaningful, then its assertion that it is notderivable is not meaningful. It is a pseudo-statement that mayappear to state something but cannot in fact state anything.For the same reason, it would be question-begging to reason thatsince the Gödel sentence states something true, its ownunderivability, it must be a genuine statement. If the sentencemakes a pseudo-statement, it does not state anything, and so cannotstate anything true. Reasoning of the sort simply assumes (as doesGödel) that the sentence is meaningful, and so fails to show that itis.In contrast, there are two compelling reasons for deeming the Gödelsentence not to be meaningful. The first of these reasons is thatany attempt to explicate the meaning of the string of symbols ofwhich the Gödel sentence is composed finds that meaning to be acomplex whole of which the meaning of that same string of symbols isa constituent. Any explanation of its meaning turns out topresuppose what it is supposed to explain. The situation differsfrom those discussed earlier in that the explanation is given interms of a string of symbols, a sentence, rather than the purportedmeaning of the symbols. The presence of a sentence creates theillusion that there is no vacuity; a statement may be vacuous but asentence is something perceptibly concrete. Nevertheless, thesituation remains essentially the same as those considered earlier.The question being asked is whether the sentence is meaningful, andthat question cannot be answered by appeal to the concreteness ofthe sentence. Such a line of reasoning would rule any string ofsymbols whatever to be meaningful. Ultimately the situation comesdown to the following: the Gödel sentence is meaningful if and onlyif the Gödel sentence is meaningful. Despite the shift fromstatement to sentence, the meaning has been given a circulardefinition, which, as we have seen, can only generate semanticvacuity and a pseudo-statement.The second reason for denying meaningfulness springs from a moregeneral consideration. The formalization of arithmetic together withits metalanguage is presumably a formalization of the arithmetic andmetalanguage that occur in natural languages, in particular, inEnglish. Its translation back into English must be possible, andmake good sense. In English, one does not speak of sentences beingtrue or of sentences being derivable, but of statements being true,and of statements being provable. The only cogent translation of theGödel sentence back into English is a statement that asserts its ownunprovability from the axioms of arithmetic and the laws of logic.Precisely such a self-evaluation of unprovability was examinedearlier and found to be a criterially deficient predication, apseudo-statement that is neither true nor false. On its intendedinterpretation, the Gödel sentence does not express a meaningfulstatement.The basic point is that for a formal system to qualify as aformalization of some discipline, it must admit of translation backinto the language of the discipline it purports to formalize. Thepoint is one that it is easy for logicians to overlook. The logicpracticed in formal systems is a form of what Husserl terms'consequence-logic' or 'logic of non-contradiction', that is, theconcern is with what follows from certain statements in accordancewith given rules, and not with the truth of the statements (HuaXVII, pp. 15-6, 58-9). In addition, as Husserl notes with regard tomathematics, it is customary for the formal system to be treatedsomewhat like a game in which strings of symbols, depending on theirform, are derivable or not derivable from other strings according torules. The signs in the system have, like chess pieces, "a gamesmeaning" that replaces the arithmetic or statemental meaning forwhich the signs are actually doing duty (Hua XVII, p. 104).Nevertheless, if the game is to allow any conclusions to be drawnabout the discipline being formalized, its strings of symbols andits rules must be interpretable, which means translatable back intothe original language. In the case of Gödel's formalization ofarithmetic, a particular sentence, the Gödel sentence, translatesinto a pseudo-statement. Such a sentence can hardly provide a soundbasis on which to build a persuasive proof of the incompleteness offormalized arithmetic.Matters are not improved if the Gödel sentence is replaced with asimpler one, one of the sort suggested by Kripke that uses a propername to refer to itself and to say that a particular sentence,itself, is not derivable. Any such sentence has nothing to do witheither arithmetic or the metalanguage of arithmetic, and so itspresence in a system of formalized arithmetic is quite unwarranted.More importantly, the definition of the name it contains iscircular. It defines the name in terms of a sentence that containsthe name, which name is not as yet a name since the point of thedefinition is to make it one. It would be no less nonsensical todeclare 'Gorg' to be a name for the word 'Gorg'--although in factthere is no such word since, prior to the definition, 'Gorg' is amere string of letters. Furthermore, the sentence in question shouldin principle be translatable back into English if it is to beconsidered a proper formalization of what it purports to formalize.On translation, the sentence becomes a nonsensical self-evaluationof unprovability. The Kripke sentence is thus no improvement on theGödel sentence.EXCERT FROM: THE LIAR SYNDROME SATS Nordic Journal of Philosophy, vol. 3, no. 1 VI. GÖDEL AND SENTENTIAL SELF-REFERENCEKurt Gödel's well-known theorem, widely termed 'Gödel'sTheorem', demonstrates that any formal system of classical two-valued logic augmented with the axioms of arithmetic and a portionof its own metalanguage will contain sentences that are undecidablein the system--sentences for which neither they nor their negationsare provable within the system. The metalinguistic evaluations aremade possible through a provability predicate defined syntacticallyas membership in the set of sentences that are immediateconsequences of the axiom-sentences. Since the provability predicateapplies to sentences rather than statements, to avoid confusion itis better termed 'a derivability predicate'. The undecidablesentence figuring in the theorem, the Gödel sentence, says that aparticular sentence, itself, is not derivable. Thus, the undecidablesentence responsible for the incompleteness apparently statessomething true, its own underivability.The paradoxical line of reasoning central to the theorem alsostrongly resembles the one found in the Liar. It runs roughly asfollows: if the sentence were derivable, it would have to be true,hence say something true, and hence, as it says, not be derivable--which contradicts the assumption of its derivability; if thenegation of the sentence were derivable, since the sentence statesits underivability, it would have to be not underivable, hencederivable--with the result that both the sentence and its negationwould be derivable. As with the Liar, each of two possiblealternatives generates a contradiction, although in the present casethe consequence is not paradox but incompleteness.The Gödel sentence figuring in Gödel's proof states that aparticular number satisfies a particular one-place propositionalfunction that defines a class of numbers. In Gödel's formal system anumber is assigned as a name to each class of numbers according toits rank in an ordering of the various classes of numbers. Roughlycharacterized, the Gödel sentence states that a particular class-number (the class-number of the class of class-numbers for which thesentences stating they possess the defining characteristics of theclasses they number are not derivable) has the definingcharacteristic of the class it numbers (that of the non-derivabilityof the sentence stating its possession of the definingcharacteristic of the class it numbers).Clearly, since the Gödel sentence, on its intended interpretation,states the underivability of a certain string of symbols, ratherthan the unprovability of what is said, it is not vulnerable to thereasoning presented earlier against statemental self-reference.Sentential self-reference is widely and plausibly esteemed to be aharmless operation. In this spirit, Saul Kripke has contended thatby interpreting elementary syntax in number theory, "Gödel put theissue of the legitimacy of self-referential sentences beyond doubt;he showed that they are as incontestably legitimate as arithmeticitself." Kripke is obviously right when 'a legitimate sentence' istaken to mean a formula of the formal system that is a well-formedformula according to the formation rules of the system. However, theimportant issue is whether such sentences are legitimate in thesense that they make good sense on their intended interpretation,rather than express dubious statements that inadequate formationrules have failed to exclude. Gödel makes no attempt to show thatthe interpreted Gödel sentence makes sense (nor does Kripke); heseems simply to assume that it makes sense given that it is well-formed according to the rules of the system. The assumption hardlycommands automatic endorsement, since, as we saw earlier withstatemental self-reference and criterially circular predication,sentences considered to be well-formed may in fact express nonsense.The issuing of a certificate of legitimacy should be contingent uponthe results of closer scrutiny of the meaning of the Gödel sentence.To clarify matters, let 'E' and 'e' represent some normal class ofnumbers (such as the class of even numbers) and its class-number,and let 'D' represent an underivability predicate. Let 'N' and 'n'represent respectively the class and class-number of all classessuch that the sentence stating that the number has the definingproperty for membership in the class it numbers, is not derivable.The necessary and sufficient conditions for each of the two class-numbers, e and n, to be members of the class of class-numbers, N,may then be stated respectively as follows:(10) Ne df D('Ee') (11) Nn df D('Nn')The statement of membership conditions in (10) is clearly notcircular. The same is not obviously the case for the statement ofmembership conditions in (11). Indeed, on further inspection, thealleged legitimacy of the Gödel sentence, the left-hand side of(11), becomes quite suspect.For instance, it might be found tempting to argue as follows infavor of the claim that Nn, the left-hand side of (11), should makeperfectly good sense. What it states is equivalent to what is statedby the right-hand side, the underivability of a particular string ofsymbols, 'Nn'. Since a string of symbols is either derivable fromthe axiom-strings or not, a statement asserting it is not derivablemust be meaningful, and hence be a genuine statement. Given theequivalence of the right-hand and left-hand statements, the Gödelsentence must also express a genuine statement. However, such a lineof reasoning begs to point at issue. The question is whether thesentence 'Nn' makes sense. If it does not, then the left-handstatement of (11) does not, and so neither does the statementequivalent to it, the right-hand side of (11). The latter must thenbe a pseudo-statement, one that appears to assert the underivabilityof a particular string of symbols, but one that in fact cannotassert anything. Thus, in assuming that the right-hand side of (11)asserts something, the argument presupposes what it purports toestablish.For the same reason, it would be fallacious to claim (as Gödeldoes) that the Gödel sentence states something true, its ownunderivability, and then to argue that since it states somethingtrue, the left-hand statement of the equivalence must also be true,and hence a genuine statement. If the sentence makes a pseudo-statement, it states nothing, and so cannot state anything true.Such an argument simply assumes (as Gödel does) that the sentencemakes a genuine statement, and so fails to show that it does.In point of fact, there are two excellent reasons for thinking thesentence cannot make a genuine statement. First, the predication onthe left-hand side of (11) is meaningful only if the statement onthe right-hand side is meaningful. The latter is meaningful only ifthe string of symbols 'Nn' is a string of symbols that expresses ameaningful statement. If the string 'Nn' expressed nonsense, thensince it is also the Gödel sentence, the latter would not make ameaningful statement. Thus, the meaningfulness of the predication,Nn, is conditional upon the meaningfulness of the statementexpressed by 'Nn', which is to say, itself. As a result, thepredication is criterially circular. The situation echoes that ofthe Grelling paradox: the attribution of a particular predicate to aparticular individual fails to make sense. In the case of the Gödelsentence the circularity is less apparent because the relevantstatement is defined in terms of its sentence rather than in termsof itself. However, the shift from statement to sentence fails toavoid circularity since the question still arises as to whether theparticular string of symbols is legitimate in the sense ofexpressing a genuine statement.The second reason for thinking the sentence illegitimate is no lessdecisive. If the formalization of arithmetic-plus-metalanguage is tobe considered a faithful rendition of arithmetic-plus-metalanguagein English, its translation back into English must make good sense.The exception could only be a situation where the formal systememploys some peculiar idiom in order to correct an incoherentEnglish one. Such appears not to be the case. It is true thatEnglish speaks of the provability of statements rather than of thederivability of sentences, but it manages to do so withoutcollapsing into incoherence. Talk of sentences being true, or false,or derivable, has its source in what is convenient for logicians,and not in the incoherence of some English idiom. In thesecircumstances, the only cogent translation of the Gödel sentenceback into English is a statement asserting its own unprovability, asin (7). Such a statement is a pseudo-statement afflicted with theLiar Syndrome, one the negative effects of which are neutralizablein English with appropriate precautions.Thus, the Gödel sentence is properly judged to be illegitimate. Itmakes a pseudo-statement, and consequently should never have beenadmitted into a formal system that is two-valued, and henceunequipped to accommodate such sentences. Moreover, since a pseudo-statement says nothing, the argument in Gödel's IncompletenessTheorem fails appealing as it does at two crucial points to what thestatement says.The theorem cannot be rescued by an appeal to the services of thesimplified version of Gödel sentence suggested by Kripke, asententially self-referential sentence constructed through the useof proper names for sentences. The definition of such a sentencemay be represented as follows, with 'n' representing a sentence name:(12) n =ds D'n'Clearly, the statement expressed by 'n' has nothing to do eitherwith arithmetic or with the metalanguage of arithmetic, so itspresence in a system of formalized arithmetic is quite unwarranted.In addition, a definition as in (12) succumbs to charges analogousto those directed above against (11). First of all, 'n' is ameaningful name of a sentence in a two-valued system only if theright-hand side of (12) is a sentence that expresses a meaningfulstatement, and the latter is the case only if the 'n' on the right-hand side is the name of a sentence that expresses a meaningfulstatement. Thus, the meaningfulness of the name 'n' has been made todepend in circular fashion upon the name 'n' being meaningful. Thesituation is not unlike that of declaring the word 'Gerg' to be aname for the word 'Gerg', whereas prior to a definition it is a merestring of letters, and not a word. Likewise, in (12) 'n' may name aname only if 'n' is already a name and hence designates something.Secondly, a formal system that is a formalization of the arithmeticand metalanguage given in a natural language should in principle betranslatable back into that language if it is to be considered aproper formalization of what it purports to formalize. Since theonly cogent translation back into English of the concept ofderivability is that of provability, the interpreted Gödel sentencebecomes a nonsensical self-evaluation of unprovability as in (9)above.Thus, the shift from statemental self-reference to sentential self-reference is, from the point of view of present concerns, of lessthan dubious utility. Statements that are self-referential andpredicates that are criterially circular in the sentential mode maybe represented as follows, where the predicate '' represents anysentential semantic predicate:(13) p =ds 'p' (14) Nn df 'Nn'(13) is, as it were, the sentential rendition of (2), while (14) isthat of (8). When transformed into their sentential correlates, thepseudo-statements that instantiate (2) and (8) become sententialevaluations that instantiate (13) and (14). Certainly, in discussingformal systems it may be useful to speak of sentences rather than ofthe statements they make, but otherwise the transformation yields nosignificant gain. If syntax faithfully reflects semantics, as itshould, the formation rules of the system must screen fordefinitions and instantiations that generate sentences expressingstatements afflicted with the Liar Syndrome. Contradiction is theprice of failure to do so.Any system that contains both semantic predicates of some sort (oftruth, provability, possibility, necessity) and names or designatorsof statements, sentences, or classes, must, if it is to avoidunnecessary problems, screen for failures of instantiation andsubstitution salva significatio. It must be suitably equipped eitherwith formation rules that eliminate any resulting nonsensical andirrelevant statements, or with a notation that prevents confusion ofthe pseudo-statements with the genuine statements that evaluatethem. The system that figures in Gödel's Theorem fails to do any ofthis.VII. IMPLICATIONSThe puzzles attendant upon self-reference have over the yearsgenerated a wide variety of extravagant claims. Although in view ofthe above findings the error of these claims is obvious enough, abrief spelling out of the obvious is perhaps not amiss.The widespread tenet that a formal language cannot contain its ownmetalanguage without generating paradox is quite overstated. It istrue only of certain formal languages, those lacking the machinerynecessary either to eliminate certain pseudo-statements or toaccommodate them in a three-valued system equipped withdisambiguators. The Liar provides no grounds to speak, as has HilaryPutnam, of "giving up the idea that we have a single unitary notionof truth applicable to any language whatsoever ... ," and hence ofgiving up any notion of a God's Eye View of the world, and embracinga general Antirealist or non-Objectivist account of human knowledge.Indeed, it would be astounding to find such claims warranted.English has been serving as its own metalanguage for an impressivelength of time without requiring the services of hermetic levels oftruth, and without collapsing into incoherence.Gödel's Theorem is often understood to show that any system offormalized arithmetic must be incomplete. In addition, it is notinfrequently touted to have other far-reaching implications. JohnStewart, for one, has argued that Gödel's Theorem undermines anObjectivist epistemology and supports transduction, the view thatsubject and object exist only in their relationship to each other.Michael Dummett deems the theorem to show "that no formal system canever succeed in embodying all the principles of proof that we shouldintuitively accept." Likewise, Roger Penrose takes it to show thatin mathematical thinking "the role of consciousness is non-algorithmic," and that "human understanding and insight cannot bereduced to any set of computational rules."As concluded above, Gödel's Theorem is made possible by a failureto either exclude or accommodate sentences that express pseudo-statements on their intended interpretation. Such a situationprovides no obvious support for the claim that mathematics has nofirm foundation, and hence none for Antifoundationalism or forAntirealism. Nor does it reveal some deep feature of mathematicalthinking, a feature that eludes capture in a formal system. Such afeature may well exist, but evidence for it must be soughtelsewhere. Finally. it cannot reasonably be claimed to reveal someremarkable capacity of the human mind: self-reference. The lattersimply generates nonsense. A capacity to lapse into nonsense,however proficiently exercised, is hardly a very awe-inspiring humantrait.

`I don't think this is serious. Take, with Smullyan the simple alphabet`

`with ~, (, ), P, N as only symbols.`

`An expression X is any non empty sequence build on that alphabet, like`

`((PNN(~~`

Assume some machine can print some expressions.

`Define a sentence to be an expression with the following shape, with X`

`being any expression (as just defined here):`

P(X) PN(X) ~P(X) ~PN(X)

`Now I give a semantic. First I define, with Smullyan, the norm of the`

`expression X to be X(X). So the norm of PP~ is PP~(PP~).`

For all expression X, I will say that:

`P(X) is true if the machine prints someday, soon or later, the`

`expression X.`

~P(X) is true if the machine never print X, PN(X) is true if the machine prints the norm of X, soon or later. ~PN(X) is true if the machine never print the norm of X.

`Smullyan asks us to assume that the machine is correct: it will never`

`print a sentence which is not true. It means that if the machine`

`prints PX some day, it will print X some (other or not) day.`

`Now, as a puzzle, Smullyan asks us to find a true sentence that the`

`machine will never prove.`

`May be you can try to find it for your own amusement, but I give the`

`solution below:`

================================

`A solution is ~PN(~PN). Indeed by the semantic, ~PN(~PN) is true only`

`if the norm of ~PN is not printable, but by the definition of the`

`norm, the norm of ~PN *is* ~PN(~PN). So ~PN(~PN) gives a simple`

`sentence, true, and not printable by any correct machine printing`

`expression in that language. The sentence affirms correctly its own`

`non printability.`

`If your argument above was correct, it should apply to this one too,`

`which is far more simple. What Gödel did consists in showing that a`

`similar argument can be made once we postulate predicate logic and`

`elementary typed set theory (his theory "Principia Mathematica").`

`Later people proved that this type of self-reference already occur for`

`very weak theory (like Robinson Arithmetic, Peano Arithmetic) and all`

`their consistent extensions. Hilbert and Bernays proved Gödel's idea`

`that Peano Arithmetica (or his PM) can prove its own incompleteness`

`theorem (and that is what I exploit).`

`It seems to me that Louis Kauffman address also very nicey this issue`

`too notably in his JPMB contribution.`

`Self-reference is not problematic, and problems occur only when people`

`confuse intensional variants of provability, or truth and provability.`

`Note also that Gödel managed to avoid the use of semantic or truth,`

`like I just did. His proof can be made simpler by using them. Today,`

`thanks to Tarski, the notion of truth is no more problematic in logic,`

`and in the elementary part of mathematics.`

Bruno Marchal ULB-IRIDIA Brussels-Belgium

`PS Note that this is my second post of this week. If you reply, I will`

`reply next week.`

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