Dear Albert,
I see that I do want to say a bit more.
Lets turn to the Godel Theorem. We have that there is a coding method that
assigns to each formula F a Godel number g that can be algorithmically decoded
into that formula.
For this I write g ——> F. Colloquially this means “g is the Godel number of the
formula F” but remember that there is an algorithm that allows this to happen
in the formal system.
Furthermore, if F(u) is a formula with a free variable u, then we would have a
g such that g ——> F(u). But u is an arithmetical variable and so we can define
a function that takes Godel numbers of this type to new Godel numbers via the
pattern
g ——> F(u)
then
#g ———> F(g).
That is, #g is the Godel number of the result of substituting g for the free
variable of F(u).
The function g to #g is arithmetically defined and can be computed by the
formal system.
Thus we can take # as an abbreviation for a special text in the formal system
and consider functions with one free variable u of the form F(#u).
Then we have
g ——> F(#u)
#g ———> F(#g).
The formula F(#g) has the Godel number #g.
I shall refrain from saying that F(#g) “talks about itself”. This is just a way
of saying that inside the formula F(#g) appears a representative for the Godel
number of the formula as a whole.
But we also have a very rich formal system, and so there is a predicate in the
system that has the form ~B(u) which is the statement that there is no proof in
the system of the formula whose Godel number is u. This is a statement that can
be made inside the formal system. Inside the system, if confronted by the
statement ~B(g) where g ——> F, the system can decode g into F and than ~B(g)
states that there is no proof of F. The formal system is equipped with enough
internal language so that such a statement is meaningful. The statement that
there is no proof of F is a statement about sequences of numbers since a proof
is a sequence of statements of a certain kind ending in F. But now, Godel uses
the # function and examines the Godel number g of ~B(#u).
We have
g ———> ~B(#u)
#g ———> ~B(#g).
So ~B(#g) asserts that there is no proof of the formula with Godel number #g
and this is ~B(#g).
The result is this:
If decoding of h is provable, then B(h) is also provable in the formal system.
For B(h) says that the decoding of h is provable.
If the formal system were to have a proof of the decoding of h then it would
also have a proof of B(h).
If the formal system were to have a proof of the decoding of #g then it would
also have a proof of B(#g).
But the decoding of #g is ~B(#g). So if there were a proof of the decoding of
#g then there would be proofs of
~B(#g) and B(#g). Hence the formal system would be inconsistent.
We assume that the formal system is consistent, and we conclude that it is
incomplete.
The point about these constructions is that they would happen all the way to
inconsistency INSIDE the formal system. It is independent of our talk from the
outside.
The Godel argument shows that if you have a sufficiently rich formal system
that includes arithmetic then there are statements in it, so that if the system
could prove them, then the system would be inconsistent. We can only conclude
that if the system is consistent then these statements have no proof in the
system.
Now finally I get to the point. We CAN interpret the statement ~B(#g) as
asserting its own
unprovability. You can unwind what it says about arithmetic. It says that there
is no valid proof in the sense of a sequence of statements in the formal system
(all encoded by numbers) so that the sequence starts with givens in the formal
system and ends with the formula
~B(#g). That means that it says that there is no proof of ~B(#g) in the system.
And we have proved that. So we have proved, from the outside that ~B(#g) is
indeed true. But only by assuming the consistency of the formal system.
Best,
Lou
> On May 9, 2016, at 1:16 AM, Albert Johnstone <bertjoh...@gmail.com> wrote:
>
> Greetings Lou,
>
> If I understand you correctly, 'P' means 'an empty string is
> printable', and so is a sentence rather than merely a predicate.
> Actually the point has no effect on Smullyan's argument, a simpler
> version of which is the following:
>
> The sentence, ''c' is not printable', is not printable by a printer
> that prints only sentences making true statements, because it contains
> the letter 'c' which, if the sentence were true, would not be
> printable.
>
> Smullyan couches his argument in a more complex form that echoes the
> situation in Gödel's theorem, and in which Smullyan's sentence echoes
> the Gödel sentence. He does so in order to suggest that the two
> sentences are essentially similar. They are not, however, because the
> Gödel sentence is supposed to express a statement in arithmetic, not a
> statement about words. As a result, Smullyan's engaging thought
> experiment ends up muddying the water more than clarifying Gödel's
> theorem.
>
> The Godel sentence says that a certain sentence is not derivable in a
> particular system. Since the system is a formalization of arithmetic,
> on the intended interpretation of the system the sentence expresses an
> arithmetic statement. However, the sentence is ambiguous in that it
> may be used to make two different statements, one a statement about
> itself, the other a statement about the statement that makes a
> statement about itself. The first is semantically self-referential,
> hence vacuous, hence neither true nor false; the other is true in that
> it says the first statement is not derivable. Since the two have
> different truth values, they have to be different statements. When
> examined phenomenologically or experientially, they are indeed quite
> different.
>
> This may seem absurd, but consider the following example:
> If someone says, 'The statement I am now making is true,' that
> statement says nothing. It is vacuous, a pseudo-statement, and so is
> neither true nor false. If a second person says that the statement is
> true, then what the second person says is false. They are both
> referring to the same statement and saying that it is true, but what
> one says has a different truth value from that of what the other says.
> This is because the first utterance is self-referential, while the
> second is not.. If they gave name 's' to the first utterance, they
> could both be using the same sentence, 's is true'.
>
> The Gödel sentence likewise makes two different statements. It is a
> possible string of symbols that makes a well-formed sentence according
> to the formation rules. Since it is a sentence in a formalized system
> of arithmetic, on its intended interpretation it makes an arithmetic
> statement. It could be used to make either of the two statements. It
> could make either a semantically self referential statement that says
> of itself that it is not provable, or it could be a
> non-self-referential statement saying that the self-referential
> statement is not provable. If it is the first, it is vacuous, hence
> neither true nor false, and cannot say anything; it only appears to
> say that it is not provable. If it is the second, it says something in
> fact true in that it says that the self-referential statement is not
> provable.
>
> What you say about consistency is true, but to my mind where Gödel
> goes wrong is both in assuming that the interpreted Gödel sentence has
> a single meaning, and in allowing his system of formalized arithmetic
> to contain sentences that make semantically self-referential
> pseudo-statements.
>
> Best regards, Bert
>
> On 5/7/16, Louis H Kauffman <lou...@gmail.com> wrote:
>> Dear Albert,
>> I cannot comment into FIS until the beginning of the week, but I want to
>> make some comments on your comments right now.
>> So will do so privately.
>>
>> You write "On reflection, however, I suspect that the sentence ‘~PR~PR’ has
>> been incorrectly interpreted. The second expression ‘~PR’ at the end of
>> Smullyan’s sentence is a well-formed formula in Smullyan’s system, but when
>> translated into English, it has no grammatical subject, and so cannot be a
>> sentence; it is merely a predicate, and so does not make a statement. Hence
>> Smullyan’s sentence must be saying that the string of symbols, ‘~PR’,
>> translatable as the predicate ‘is not printable’, is not printable.”
>>
>> The empty string is a valid predicate.
>> ~PR by the rules is meaningful, and it says that the repetition of the empty
>> string (which is just the empty string) is not printable.
>> That is it says that when you push the button on the machine the machine
>> must print some characters.
>> Of course if the machine does not print ~PR, then this statement is not
>> known to be true or false. But if on pressing the machine’s button you
>> receive the string ~PR, then you can be assured that the machine will never
>> print an empty string.
>>
>> Now consider ~PRX for some string X. This string is meaningful and it
>> asserts that the machine can never print XX. The meaning of X (if it has
>> one) is irrelevant. The only meaning of
>> ~PRX is in reference to printing XX.
>> Thus if you found the machine had printed ~PRPPP,
>> then this would mean that the machine would never print PPPPPP.
>>
>> By the same token, if the machine prints ~PR~PR then it cannot print the
>> repetition of ~PR but this repetition is ~PR~PR and so the machine cannot
>> print ~PR~PR.
>> If printed, the string ~PR~PR asserts that it cannot be printed.
>> If the machine should print ~PR~PR then it would be inconsistent, stating
>> that it could not print ~PR~PR.
>> We have assumed that the machine is consistent and therefore it will not
>> print this string.
>> So we see that the assumption of consistency (that the machine always tells
>> the truth when printing a meaningful string) allows us, as observers of the
>> machine, to conclude that the sentence ~PR~PR is true and so there are
>> sentences that the machine cannot print that are nevertheless true.
>>
>> The interesting thing about Smullyan’s model is that it embodies most of the
>> aspects of Godels’s theorem that are often under philosophical and semantic
>> discussion.
>> The key point about Godel’s Theorem is that the formal system must be
>> assumed to be consistent. Without this we can conclude nothing.
>>
>> Note that this illustrates the fact that it is wrong to say that Godel’s
>> Theorem shows that we can prove theorems that machines cannot prove. Such a
>> statement is based on the assumption
>> that a proving machine can be built that is consistent. We do not know that.
>> But it does show that we can prove theorems that consistent machines cannot
>> prove.
>> Who would be interested in a machine that was not consistent?
>> Best,
>> Lou
>>
>>
>>> On May 6, 2016, at 11:57 PM, Albert A Johnstone <alb...@uoregon.edu>
>>> wrote:
>>>
>>> Greetings everyone,
>>> I’d like to say a few words about Smullyan’s thought experiment and its
>>> relevance to Gödel’s Theorem in the hope of putting an end to discussion
>>> of a topic somewhat tangential to the main one. Before doing so, I am
>>> forwarding an email from Lou Kauffman which gives a very clear account of
>>> Smullyan’s reasoning.
>>>
>>> -------- Original Message --------
>>> Subject: Re: [Fis] _ FIS discussion
>>> Date: 2016-05-04 12:30
>>> From: Louis H Kauffman <lou...@gmail.com>
>>> To: Maxine Sheets-Johnstone <m...@uoregon.edu>
>>>
>>> Dear Maxine,
>>> I am writing privately to you since I have used up my quota of forum
>>> comments for this week.
>>> I am going to discuss a Smullyan puzzle in detail with you.
>>> I call this the Smullyan Machine.
>>>
>>> THE SMULLYAN MACHINE
>>> The machine has a button on the top and when you press that button, it
>>> prints a string of symbols using the following three letter alphabet.
>>> { P, ~ ,R}
>>> Thus the machine might print P~~~NRRP.
>>> I shall designate an unknown string of symbols by X or Y.
>>> Strings that begin with P, ~P, PR or ~PR are INTERPRETED (given meaning)
>>> as follows:
>>>
>>> Meaningful Strings
>>> (When I say “X can be printed by the Machine” I mean that when you press
>>> the button the machine will print exactly X and nothing else.)
>>>
>>> PX: X can be printed by the Machine.
>>> ~PX: X cannot be printed by the Machine.
>>> PRX: XX can be printed by the Machine.
>>> ~PRX: XX can not be printed by the Machine.
>>>
>>> Thus it is possible that the machine might print
>>> ~PPR
>>> This has meaning and it states that the machine cannot bring PR all by
>>> itself when the button is pressed.
>>>
>>> AXIOM OF THE MACHINE
>>> The Smullyan Machine always tells the truth when it prints a meaningful
>>> string.
>>>
>>> THEOREM. There is a meaningful string that is true but not printable by
>>> the Smullyan Machine.
>>>
>>> PROOF. Let S = ~PR~PR. This string is meaningful since it starts with
>>> ~PR.
>>> Note that S = ~PRX where X = ~PR. Thus by the definition (above) of the
>>> meaning of S, “XX is not printable by the Machine.”
>>> We note however that XX = ~PR~PR = S. Thus S has the meaning that “S is
>>> not printable by the Machine.”
>>> Since the Machine always tells the truth, it would be in a contradiction
>>> if it printed S. Therefore the Machine cannot print S.
>>> But this is exactly the meaning of S, and so S is true. S is a true but
>>> not printable string. The completes the proof.
>>> —————————————————————————————————————————————————————
>>>
>>> Now I have an assignment for you.
>>> Please criticize the Smullyan Machine from your phenomenological point of
>>> view.
>>> If you wish you could include my description of the Machine and make a
>>> statement about it on FIS.
>>> My point and Smullyan’s point in his Oxford University Press Book on
>>> Godel’s Theorem, is that the Machine is an accurate depiction of the Godel
>>> argument, with
>>> Printabilty replacing Provablity. The way that self-reference works here,
>>> and the way the semantics and syntax are controlled is very much like the
>>> way these things happen in the
>>> full Godel theorem. The Machine provides a microcosm for the discussion of
>>> Godel and self-reference.
>>> Yours truly,
>>> Lou Kauffman
>>> P.S. “This sentence has thirty-three letters.”
>>> is a fully meaningful and true English sentence.
>>> Self-referential sentence can have meaning and reference.
>>> ____________________________________________________________________
>>>
>>> Johnstone again:
>>>
>>> In response to the above assessment, let us first distinguish syntactic
>>> self-reference which is reference to the words or sentence that one is
>>> using, from semantic self-reference, which is reference to the MEANING of
>>> the words or sentences one is using. There is nothing wrong with syntactic
>>> self-reference but semantic self-reference invariably generates vacuity
>>> and sometimes paradox.
>>>
>>> Now Smullyan’s sentence ‘~PR~PR’ is often interpreted (as by Lou, Bruno,
>>> and by myself earlier) as making a syntactically self-referential
>>> statement that says that the sentence expressing that statement is not
>>> printable. On the supposition that such is the case, the statement it
>>> makes must also be semantically self-referential for the following reason.
>>> In Smullyan’s scenario, the printing machine prints only true statements.
>>> As a result, a sentence is printable if and only if the statement it makes
>>> is true. Consequently, the two predicates ‘is not printable’ and ‘is not
>>> true’ are logically equivalent. A sentence that says of itself that it is
>>> not printable is consequently logically equivalent (each entails the
>>> other) to a statement that says of itself that it is not true, that is, it
>>> is equivalent to a Liar statement. As such, it is semantically incomplete
>>> or vacuous; it does not make a statement, and hence is neither true nor
>>> false, and so cannot possibly be an unprintable true statement.
>>> The equivalence of the two predicates has the result that ‘~PR~PR’ is both
>>> syntactically AND semantically self-referential.
>>>
>>> On reflection, however, I suspect that the sentence ‘~PR~PR’ has been
>>> incorrectly interpreted. The second expression ‘~PR’ at the end of
>>> Smullyan’s sentence is a well-formed formula in Smullyan’s system, but
>>> when translated into English, it has no grammatical subject, and so cannot
>>> be a sentence; it is merely a predicate, and so does not make a statement.
>>> Hence Smullyan’s sentence must be saying that the string of symbols,
>>> ‘~PR’, translatable as the predicate ‘is not printable’, is not
>>> printable.
>>>
>>> On this second interpretation of the Smullyan sentence, ‘~PR~PR’ is still
>>> a sentence that cannot be printed by a machine that prints only strings of
>>> symbols that make true statements. This is because, on one hand, if what
>>> the sentence says is true, then it is true that ‘~PR’ is unprintable;
>>> however, since the sentence itself contains that string of words, it
>>> cannot be printed. On the other hand, if what the sentence says is false,
>>> it cannot be printed because the printer prints only what is true. The
>>> Smullyan sentence, whether the statement it makes is true or a false,
>>> cannot be printed by a printer that prints only sentences that make true
>>> statements. It could, of course, be printed by a different printer, one
>>> that also prints false statements such as it.
>>>
>>> On this second interpretation of ‘~PR~PR’, the Gödel sentence differs from
>>> Smullyan’s sentence in that its subject is a sentence, not a predicate. It
>>> states that a certain sentence, itself, is not provable in a certain
>>> formal system. The sentence allegedly makes an arithmetical statement on
>>> its intended interpretation, but since it is semantically self-referential
>>> (like the statement that this statement is true) it is vacuous and so says
>>> nothing, much less something that is true. Because it says nothing, it has
>>> no business being in a system of formalized arithmetic.
>>> _______________________________________________
>>> Fis mailing list
>>> Fis@listas.unizar.es
>>> http://listas.unizar.es/cgi-bin/mailman/listinfo/fis
>>
>>
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