[Fis] FIS Discussion (No Vol #)
Bruno Marchal
marchal at ulb.ac.be
Mon May 2 20:03:33 CEST 2016
On 02 May 2016, at 03:38, Maxine Sheets-Johnstone wrote:
> To all concerned colleagues,
>
> I appreciate the fact that discussions should be conversations about
> issues,
> but this particular issue and in particular the critique cited in my
> posting
> warrant extended exposition in order to show the reasoning upholding
> the critique.
> I am thus quoting from specific articles, the first
> phenomenological, the second
> analytic-logical--though they are obviously complementary as befits
> discussions
> in 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. Johnstone
>
> The aim of this article is to show that a Husserlian approach to the
> Liar paradoxes and to their closely related kin discloses the
> illusory nature of these difficulties. Phenomenological meaning
> analysis finds the ultimate source of mischief to be circular
> definition, implicit or explicit. Definitional circularity lies at
> the root both of the self-reference integral to the statements that
> generate Liar paradoxes, and of the particular instances of
> predicate criteria featured in the Grelling paradox as well as in
> the self-evaluating Gödel sentence crucial to Gödel's theorem.
> Since the statements thereby generated turn out on closer scrutiny
> to be vacuous and semantically nonsensical, their rejection from
> reasonable discourse is both warranted and imperative. Naturally
> enough, their exclusion dissolves the various problems created by
> their presence. . . .
>
> VII: THE GOEDEL SENTENCE
> Following a procedure invented by Gödel, one may assign numbers in
> some orderly way as names or class-numbers to each of the various
> classes of numbers (the prime numbers, the odd numbers, and so on).
> Some of these class-numbers will qualify for membership in the class
> they name; others will not. For instance, if the number 41 should
> happen to be the class-number that names the class of numbers that
> are divisible by 7, then since 41 does not have the property of
> being divisible by 7, the class-number 41 would not be a member of
> the class it names.
> Now, consider the class-number of the class of class-numbers that
> are members of the class they name. Does it have the defining
> property of the class it names? The question is unanswerable. Since
> the defining property of the class is that of being a class-number
> that is a member of the class it names, the necessary and sufficient
> condition for the class-number in question to be a member of the
> class it names turns out to be that it be a member of the class it
> names. In short, the number is a member if and only if it is a
> member. The criterion is circular--defined in terms of what was to
> be defined--and consequently not a criterion at all since it
> provides no way of determining whether or not the number is a member.
> The situation is obviously similar for the class-number of the
> complementary class of class-numbers--those that do not have the
> defining property of the class they name--since the criteria in the
> two cases are logically interdependent. The criterion of membership
> is likewise defined in circular fashion, and hence is vacuous. In
> addition, the criterion postulates an absurd analytic equivalence,
> that of the defining property with its negative. The question of
> whether the class-number is a member of the class it names is
> unanswerable, with the result that any proposed answer is neither
> true nor false. In addition, of course, any answer would generate
> paradox: the number has the requisite defining property if and only
> if it does not have it.
> As might be expected, the situation is not significantly different
> for the class-number of classes of which the definition involves
> semantic predicates. Consider, for instance, the class of class-
> numbers of which it is provable that they are members of the class
> they name. The question of whether the class-number of the class is
> a member of the class it numbers is undecidable. The possession by
> the class-number of the property requisite for membership is
> conditional upon the question of whether it provably possesses the
> property, with the result that the question can have no answer.
> Otherwise stated, the number has the defining property of the class
> it names if and only if it provably has that property. In these
> circumstances, the explanation of what it means for the class-number
> to have the property has to be circular in that it must define
> having the property in terms of having the property. The vacuity
> that results is hidden somewhat by the presence of the requirement
> of provability, but while provability might count as a necessary
> condition, in the present case it cannot be a sufficient one. In
> fact, its presence creates a semantically absurd situation: the
> analytic equivalence of having the property and provably having it.
> The statement of the possession of the property by the class-number
> in question is consequently both vacuous and semantically absurd,
> hence an undecidable pseudo-statement.
> The analytic equivalence of the number's having the property and
> provably having it has a further and quite interesting consequence.
> In principle, since the equivalence is analytic, it explains what it
> means to say that the class-number in question has the requisite
> property, that is, it explains what is being said by the statement
> that attributes the property to the number. What the statement is
> saying, according to the equivalence, is that it is provable that
> the number has the property, which is to say, it is saying of itself
> that it is provable. Thus, the statement is self-evaluating. It is
> not, strictly speaking, self-referential since it contains no
> designator, and so cannot refer to itself. However, it mirrors the
> self-referential statements of the sort discussed earlier in that it
> predicates a semantic property of itself (or at least purports to do
> so).
> In these circumstances, it is not overly surprising to find that a
> sentence having a vacuously defined semantic predicate of
> provability is ambiguous or leads a double life. It may be used to
> express either of two statements, a pseudo-statement that purports
> to evaluate itself as provable, or, a genuine statement that
> evaluates the pseudo-statement, which genuine statement is, of
> course, 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 the
> pseudo-statement, the statement itself (that a particular number has
> a 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 classes
> involving semantic predicates. If the term 'heterological' that
> figures in the Grelling Paradox were defined as applying to those
> words of which it is false that they are heterological, then the
> resulting Grelling statement (the statement that 'heterological' is
> heterological) could be plausibly interpreted to be self-evaluating.
> It would be analytically equivalent to the statement that it is
> false that 'heterological' is heterological--an equivalence that may
> be read as saying that the Grelling statement says of itself that it
> is false. This second statement would, of course, find itself
> expressed by a sentence that leads a double life.
> Of particular interest for the purpose of understanding the error
> that invalidates Gödel's theorem is the case of the class-number
> that names the class of class-numbers that are not provably members
> of the class they name. Once again, the question as to whether the
> class-number that names this class is a member of the class it names
> is unanswerable. The statement that the class-number possesses the
> required defining characteristic is a criterially deficient
> predication, and hence a pseudo-statement. In addition, the
> statement 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. It
> is thus self-evaluating, and when stated in this form, it is
> expressed by a sentence that leads a double life. As a result, any
> formal system that admits and purports to accommodate a criterially
> deficient predication of the sort will also require the elaborate
> supplementary machinery found necessary to accommodate self-
> referential statements: a three-valued logic, a procedure for
> determining which instantiations of predicates (or substitutions
> into propositional functions) produce pseudo-statements, and some
> notational device for distinguishing pseudo-statements from the
> genuine statements that are their sentential doubles. As we shall
> now see, in view of the similarity in structure of the above
> statement to the Gödel sentence, analogous remarks apply to the
> latter.
> VII. THE GÖDEL SENTENCE
> In his well-known theorem Kurt Gödel purports to show that any
> formal system of classical logic equivalent to that of Principia
> Mathematica to which arithmetic constants and the axioms of
> arithmetic (Peano's) have been added, will contain sentences that
> are undecidable--that is, sentences such that neither they nor their
> negations are provable within the system. To this end he introduces
> a provability predicate defined syntactically as membership in the
> set of sentences that are immediate consequences of the axiom-
> sentences. Since the provability predicate applies to sentences
> rather than statements, to avoid confusion it is best termed 'a
> derivability predicate'. As in the arguments of the previous
> section, Gödel has a number assigned as a name to each class of
> numbers according to its rank in an ordering of the various classes
> of numbers. Roughly characterized, the undecidable sentence figuring
> in the theorem (the Gödel sentence) states that a particular class-
> number satisfies a particular one-place propositional function that
> defines a class of numbers. A little more precisely, it states that
> a particular class-number has the defining characteristic of the
> class it numbers, which class is the class of class-numbers such
> that the sentences stating that the class-numbers possess the
> defining characteristics of the classes they name are not
> derivable. In his informal introduction to his theorem, Gödel
> points out that the sentence may be read as stating via its Gödel
> number that a particular sentence, itself, is not derivable.
> The crucial line of reasoning in the theorem strongly resembles the
> one found in the Liar. It runs roughly as follows: 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 the negation of the sentence were
> derivable, then since the sentence states its underivability, it
> would have to be not underivable, hence derivable--with the result
> that both the sentence and its negation would be derivable, a
> contradiction. As with the Liar, each of two possible alternatives
> generates a contradiction, although in the present case the
> consequence is not paradox but undecidability-- undecidability in
> the form of a sentence of which neither its truth nor its falsity is
> derivable in the system. Gödel reasons that since the undecidable
> sentence apparently states something true, its own underivability,
> the system contains underivable true sentences, and hence is
> incomplete.
> The Gödel sentence is concerned with derivability rather than
> provability, or sentences rather than statements. As a result one
> may plausibly question whether it is vulnerable to the criticisms
> directed above against criterially circular predications and self-
> evaluations. While the Gödel sentence clearly differs from the
> latter, it is possible nevertheless to raise the question of its
> legitimacy. Gödel himself simply assumes that the sentence is
> legitimate--which, of course, it is in the narrow sense that it
> conforms to the formation rules of the system in which it figures.
> However, it does not follow that it is legitimate in the broader
> sense that the interpreted sentence makes sense. As we saw earlier
> with self-referential statements and criterially circular
> predications, sentences that are apparently well-formed may in fact
> express nonsense. The Gödel sentence may well express just such a
> pseudo-statement, and have nevertheless been admitted into the
> formal system through an inadequacy of the formation rules. Gödel
> dismisses the possibility of faulty circularity on the grounds that
> the sentence states only that a certain well-defined formula is
> unprovable, which formula turns out after the fact to be the one
> that expresses the proposition itself. Yet, an answer of the sort
> will not do. Where circularity results from a substitution, being
> adventitious and well-formed according to the rules do nothing to
> remove the circularity. A statement with a circularly defined
> predicate is semantically vacuous, and hence not a genuine
> statement. Thus, the question of the meaning of the sentence, the
> statement it expresses, calls for serious examination.
> A first rather curious fact that more careful scrutiny brings to
> light is that the most obvious reasons for thinking the sentence
> meaningful are actually inconclusive. For instance, it might be
> found tempting to argue as follows: that any particular string of
> symbols is either derivable from the axiom-strings or not, and hence
> since the Gödel sentence asserts that a particular string is not
> derivable, whether true or not, it must at least be meaningful.
> However, the reasoning begs the point at issue. If the Gödel
> sentence is not meaningful, then its assertion that it is not
> derivable is not meaningful. It is a pseudo-statement that may
> appear to state something but cannot in fact state anything.
> For the same reason, it would be question-begging to reason that
> since the Gödel sentence states something true, its own
> underivability, it must be a genuine statement. If the sentence
> makes a pseudo-statement, it does not state anything, and so cannot
> state anything true. Reasoning of the sort simply assumes (as does
> Gödel) that the sentence is meaningful, and so fails to show that it
> is.
> In contrast, there are two compelling reasons for deeming the Gödel
> sentence not to be meaningful. The first of these reasons is that
> any attempt to explicate the meaning of the string of symbols of
> which the Gödel sentence is composed finds that meaning to be a
> complex whole of which the meaning of that same string of symbols is
> a constituent. Any explanation of its meaning turns out to
> presuppose what it is supposed to explain. The situation differs
> from those discussed earlier in that the explanation is given in
> terms of a string of symbols, a sentence, rather than the purported
> meaning of the symbols. The presence of a sentence creates the
> illusion that there is no vacuity; a statement may be vacuous but a
> sentence is something perceptibly concrete. Nevertheless, the
> situation remains essentially the same as those considered earlier.
> The question being asked is whether the sentence is meaningful, and
> that question cannot be answered by appeal to the concreteness of
> the sentence. Such a line of reasoning would rule any string of
> symbols whatever to be meaningful. Ultimately the situation comes
> down to the following: the Gödel sentence is meaningful if and only
> if the Gödel sentence is meaningful. Despite the shift from
> statement to sentence, the meaning has been given a circular
> definition, which, as we have seen, can only generate semantic
> vacuity and a pseudo-statement.
> The second reason for denying meaningfulness springs from a more
> general consideration. The formalization of arithmetic together with
> its metalanguage is presumably a formalization of the arithmetic and
> metalanguage that occur in natural languages, in particular, in
> English. Its translation back into English must be possible, and
> make good sense. In English, one does not speak of sentences being
> true or of sentences being derivable, but of statements being true,
> and of statements being provable. The only cogent translation of the
> Gödel sentence back into English is a statement that asserts its own
> unprovability from the axioms of arithmetic and the laws of logic.
> Precisely such a self-evaluation of unprovability was examined
> earlier and found to be a criterially deficient predication, a
> pseudo-statement that is neither true nor false. On its intended
> interpretation, the Gödel sentence does not express a meaningful
> statement.
> The basic point is that for a formal system to qualify as a
> formalization of some discipline, it must admit of translation back
> into the language of the discipline it purports to formalize. The
> point is one that it is easy for logicians to overlook. The logic
> practiced in formal systems is a form of what Husserl terms
> 'consequence-logic' or 'logic of non-contradiction', that is, the
> concern is with what follows from certain statements in accordance
> with given rules, and not with the truth of the statements (Hua
> XVII, pp. 15-6, 58-9). In addition, as Husserl notes with regard to
> mathematics, it is customary for the formal system to be treated
> somewhat like a game in which strings of symbols, depending on their
> form, are derivable or not derivable from other strings according to
> rules. The signs in the system have, like chess pieces, "a games
> meaning" that replaces the arithmetic or statemental meaning for
> which the signs are actually doing duty (Hua XVII, p. 104).
> Nevertheless, if the game is to allow any conclusions to be drawn
> about the discipline being formalized, its strings of symbols and
> its rules must be interpretable, which means translatable back into
> the original language. In the case of Gödel's formalization of
> arithmetic, a particular sentence, the Gödel sentence, translates
> into a pseudo-statement. Such a sentence can hardly provide a sound
> basis on which to build a persuasive proof of the incompleteness of
> formalized arithmetic.
> Matters are not improved if the Gödel sentence is replaced with a
> simpler one, one of the sort suggested by Kripke that uses a proper
> name to refer to itself and to say that a particular sentence,
> itself, is not derivable. Any such sentence has nothing to do with
> either arithmetic or the metalanguage of arithmetic, and so its
> presence in a system of formalized arithmetic is quite unwarranted.
> More importantly, the definition of the name it contains is
> circular. It defines the name in terms of a sentence that contains
> the name, which name is not as yet a name since the point of the
> definition is to make it one. It would be no less nonsensical to
> declare 'Gorg' to be a name for the word 'Gorg'--although in fact
> there is no such word since, prior to the definition, 'Gorg' is a
> mere string of letters. Furthermore, the sentence in question should
> in principle be translatable back into English if it is to be
> considered a proper formalization of what it purports to formalize.
> On translation, the sentence becomes a nonsensical self-evaluation
> of unprovability. The Kripke sentence is thus no improvement on the
> Gödel sentence.
>
> EXCERT FROM:
> THE LIAR SYNDROME
> SATS Nordic Journal of Philosophy, vol. 3, no. 1
>
> VI. GÖDEL AND SENTENTIAL SELF-REFERENCE
> Kurt Gödel's well-known theorem, widely termed 'Gödel's
> Theorem', demonstrates that any formal system of classical two-
> valued logic augmented with the axioms of arithmetic and a portion
> of its own metalanguage will contain sentences that are undecidable
> in the system--sentences for which neither they nor their negations
> are provable within the system. The metalinguistic evaluations are
> made possible through a provability predicate defined syntactically
> as membership in the set of sentences that are immediate
> consequences of the axiom-sentences. Since the provability predicate
> applies to sentences rather than statements, to avoid confusion it
> is better termed 'a derivability predicate'. The undecidable
> sentence figuring in the theorem, the Gödel sentence, says that a
> particular sentence, itself, is not derivable. Thus, the undecidable
> sentence responsible for the incompleteness apparently states
> something true, its own underivability.
> The paradoxical line of reasoning central to the theorem also
> strongly resembles the one found in the Liar. It runs roughly as
> follows: 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 the
> negation of the sentence were derivable, since the sentence states
> its underivability, it would have to be not underivable, hence
> derivable--with the result that both the sentence and its negation
> would be derivable. As with the Liar, each of two possible
> alternatives generates a contradiction, although in the present case
> the consequence is not paradox but incompleteness.
> The Gödel sentence figuring in Gödel's proof states that a
> particular number satisfies a particular one-place propositional
> function that defines a class of numbers. In Gödel's formal system a
> number is assigned as a name to each class of numbers according to
> its rank in an ordering of the various classes of numbers. Roughly
> characterized, the Gödel sentence states that a particular class-
> number (the class-number of the class of class-numbers for which the
> sentences stating they possess the defining characteristics of the
> classes they number are not derivable) has the defining
> characteristic of the class it numbers (that of the non-derivability
> of the sentence stating its possession of the defining
> characteristic of the class it numbers).
> Clearly, since the Gödel sentence, on its intended interpretation,
> states the underivability of a certain string of symbols, rather
> than the unprovability of what is said, it is not vulnerable to the
> reasoning presented earlier against statemental self-reference.
> Sentential self-reference is widely and plausibly esteemed to be a
> harmless operation. In this spirit, Saul Kripke has contended that
> by interpreting elementary syntax in number theory, "Gödel put the
> issue of the legitimacy of self-referential sentences beyond doubt;
> he showed that they are as incontestably legitimate as arithmetic
> itself." Kripke is obviously right when 'a legitimate sentence' is
> taken to mean a formula of the formal system that is a well-formed
> formula according to the formation rules of the system. However, the
> important issue is whether such sentences are legitimate in the
> sense that they make good sense on their intended interpretation,
> rather than express dubious statements that inadequate formation
> rules have failed to exclude. Gödel makes no attempt to show that
> the interpreted Gödel sentence makes sense (nor does Kripke); he
> seems simply to assume that it makes sense given that it is well-
> formed according to the rules of the system. The assumption hardly
> commands automatic endorsement, since, as we saw earlier with
> statemental 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 upon
> the results of closer scrutiny of the meaning of the Gödel sentence.
> To clarify matters, let 'E' and 'e' represent some normal class of
> numbers (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 classes
> such that the sentence stating that the number has the defining
> property 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 not
> circular. The same is not obviously the case for the statement of
> membership conditions in (11). Indeed, on further inspection, the
> alleged 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 in
> favor of the claim that Nn, the left-hand side of (11), should make
> perfectly good sense. What it states is equivalent to what is stated
> by the right-hand side, the underivability of a particular string of
> symbols, 'Nn'. Since a string of symbols is either derivable from
> the axiom-strings or not, a statement asserting it is not derivable
> must be meaningful, and hence be a genuine statement. Given the
> equivalence of the right-hand and left-hand statements, the Gödel
> sentence must also express a genuine statement. However, such a line
> of reasoning begs to point at issue. The question is whether the
> sentence 'Nn' makes sense. If it does not, then the left-hand
> statement of (11) does not, and so neither does the statement
> equivalent to it, the right-hand side of (11). The latter must then
> be a pseudo-statement, one that appears to assert the underivability
> of a particular string of symbols, but one that in fact cannot
> assert anything. Thus, in assuming that the right-hand side of (11)
> asserts something, the argument presupposes what it purports to
> establish.
> For the same reason, it would be fallacious to claim (as Gödel
> does) that the Gödel sentence states something true, its own
> underivability, and then to argue that since it states something
> true, 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 sentence
> makes a genuine statement, and so fails to show that it does.
> In point of fact, there are two excellent reasons for thinking the
> sentence cannot make a genuine statement. First, the predication on
> the left-hand side of (11) is meaningful only if the statement on
> the right-hand side is meaningful. The latter is meaningful only if
> the string of symbols 'Nn' is a string of symbols that expresses a
> meaningful statement. If the string 'Nn' expressed nonsense, then
> since it is also the Gödel sentence, the latter would not make a
> meaningful statement. Thus, the meaningfulness of the predication,
> Nn, is conditional upon the meaningfulness of the statement
> expressed by 'Nn', which is to say, itself. As a result, the
> predication is criterially circular. The situation echoes that of
> the Grelling paradox: the attribution of a particular predicate to a
> particular individual fails to make sense. In the case of the Gödel
> sentence the circularity is less apparent because the relevant
> statement is defined in terms of its sentence rather than in terms
> of itself. However, the shift from statement to sentence fails to
> avoid circularity since the question still arises as to whether the
> particular string of symbols is legitimate in the sense of
> expressing a genuine statement.
> The second reason for thinking the sentence illegitimate is no less
> decisive. If the formalization of arithmetic-plus-metalanguage is to
> be considered a faithful rendition of arithmetic-plus-metalanguage
> in English, its translation back into English must make good sense.
> The exception could only be a situation where the formal system
> employs some peculiar idiom in order to correct an incoherent
> English one. Such appears not to be the case. It is true that
> English speaks of the provability of statements rather than of the
> derivability of sentences, but it manages to do so without
> collapsing 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 these
> circumstances, the only cogent translation of the Gödel sentence
> back into English is a statement asserting its own unprovability, as
> in (7). Such a statement is a pseudo-statement afflicted with the
> Liar Syndrome, one the negative effects of which are neutralizable
> in English with appropriate precautions.
> Thus, the Gödel sentence is properly judged to be illegitimate. It
> makes a pseudo-statement, and consequently should never have been
> admitted into a formal system that is two-valued, and hence
> unequipped to accommodate such sentences. Moreover, since a pseudo-
> statement says nothing, the argument in Gödel's Incompleteness
> Theorem fails appealing as it does at two crucial points to what the
> statement says.
> The theorem cannot be rescued by an appeal to the services of the
> simplified version of Gödel sentence suggested by Kripke, a
> sententially self-referential sentence constructed through the use
> of proper names for sentences. The definition of such a sentence
> may 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 either
> with arithmetic or with the metalanguage of arithmetic, so its
> presence in a system of formalized arithmetic is quite unwarranted.
> In addition, a definition as in (12) succumbs to charges analogous
> to those directed above against (11). First of all, 'n' is a
> meaningful name of a sentence in a two-valued system only if the
> right-hand side of (12) is a sentence that expresses a meaningful
> statement, and the latter is the case only if the 'n' on the right-
> hand side is the name of a sentence that expresses a meaningful
> statement. Thus, the meaningfulness of the name 'n' has been made to
> depend in circular fashion upon the name 'n' being meaningful. The
> situation is not unlike that of declaring the word 'Gerg' to be a
> name for the word 'Gerg', whereas prior to a definition it is a mere
> string of letters, and not a word. Likewise, in (12) 'n' may name a
> name only if 'n' is already a name and hence designates something.
> Secondly, a formal system that is a formalization of the arithmetic
> and metalanguage given in a natural language should in principle be
> translatable back into that language if it is to be considered a
> proper formalization of what it purports to formalize. Since the
> only cogent translation back into English of the concept of
> derivability is that of provability, the interpreted Gödel sentence
> becomes 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 less
> than dubious utility. Statements that are self-referential and
> predicates that are criterially circular in the sentential mode may
> be represented as follows, where the predicate '' represents any
> sentential semantic predicate:
> (13) p =ds 'p'
> (14) Nn df 'Nn'
> (13) is, as it were, the sentential rendition of (2), while (14) is
> that of (8). When transformed into their sentential correlates, the
> pseudo-statements that instantiate (2) and (8) become sentential
> evaluations that instantiate (13) and (14). Certainly, in discussing
> formal systems it may be useful to speak of sentences rather than of
> the statements they make, but otherwise the transformation yields no
> significant gain. If syntax faithfully reflects semantics, as it
> should, the formation rules of the system must screen for
> definitions and instantiations that generate sentences expressing
> statements afflicted with the Liar Syndrome. Contradiction is the
> price of failure to do so.
> Any system that contains both semantic predicates of some sort (of
> truth, provability, possibility, necessity) and names or designators
> of statements, sentences, or classes, must, if it is to avoid
> unnecessary problems, screen for failures of instantiation and
> substitution salva significatio. It must be suitably equipped either
> with formation rules that eliminate any resulting nonsensical and
> irrelevant statements, or with a notation that prevents confusion of
> the pseudo-statements with the genuine statements that evaluate
> them. The system that figures in Gödel's Theorem fails to do any of
> this.
> VII. IMPLICATIONS
> The puzzles attendant upon self-reference have over the years
> generated a wide variety of extravagant claims. Although in view of
> the above findings the error of these claims is obvious enough, a
> brief spelling out of the obvious is perhaps not amiss.
> The widespread tenet that a formal language cannot contain its own
> metalanguage without generating paradox is quite overstated. It is
> true only of certain formal languages, those lacking the machinery
> necessary either to eliminate certain pseudo-statements or to
> accommodate them in a three-valued system equipped with
> disambiguators. The Liar provides no grounds to speak, as has Hilary
> Putnam, of "giving up the idea that we have a single unitary notion
> of truth applicable to any language whatsoever ... ," and hence of
> giving up any notion of a God's Eye View of the world, and embracing
> a 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 impressive
> length of time without requiring the services of hermetic levels of
> truth, and without collapsing into incoherence.
> Gödel's Theorem is often understood to show that any system of
> formalized arithmetic must be incomplete. In addition, it is not
> infrequently touted to have other far-reaching implications. John
> Stewart, for one, has argued that Gödel's Theorem undermines an
> Objectivist epistemology and supports transduction, the view that
> subject and object exist only in their relationship to each other.
> Michael Dummett deems the theorem to show "that no formal system can
> ever succeed in embodying all the principles of proof that we should
> intuitively accept." Likewise, Roger Penrose takes it to show that
> in mathematical thinking "the role of consciousness is non-
> algorithmic," and that "human understanding and insight cannot be
> reduced to any set of computational rules."
> As concluded above, Gödel's Theorem is made possible by a failure
> to either exclude or accommodate sentences that express pseudo-
> statements on their intended interpretation. Such a situation
> provides no obvious support for the claim that mathematics has no
> firm foundation, and hence none for Antifoundationalism or for
> Antirealism. Nor does it reveal some deep feature of mathematical
> thinking, a feature that eludes capture in a formal system. Such a
> feature may well exist, but evidence for it must be sought
> elsewhere. Finally. it cannot reasonably be claimed to reveal some
> remarkable capacity of the human mind: self-reference. The latter
> simply generates nonsense. A capacity to lapse into nonsense,
> however proficiently exercised, is hardly a very awe-inspiring human
> trait.
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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