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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