[Fis] Gödel discussion
Bruno Marchal
marchal at ulb.ac.be
Tue May 10 15:14:16 CEST 2016
Dear Albert,
On 07 May 2016, at 06:57, Albert A Johnstone 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 <loukau at gmail.com>
> To: Maxine Sheets-Johnstone <msj at 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.)
Actually Smullyan meant "printable" or "printed soon or later by a
machine which is programmed to print all what she can print (it can be
shown easily that this is always possible by a dovetailing technic).
The point will be that if the machine is correct, then the set of what
is printable will be included properly in the set of what is true.
>
> 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.
It follows only that all the sentences printed by the machine will be
true. It does not entail that the machine will print all true
sentences. It is "only if", not "if and only if" when you say "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.
Which of course they can't, but Smullyan's assumption is that the
machine is correct. Not that the machine is complete (prove all true
sentences).
> 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 whole point of Gödel is to use "provable" instead of true. As you
notice below, but see Lou comment on this.
If we use the notion of truth, the argument will show that "truth",
contrary to "provable" is not even definable or expressible in the
language of the theory. That is Tarki's theorem.
Gödel proved a famous diagonal lemma: for all definable predicate P(x)
it exists a sentence K such that
PA proves K <-> P("K"), where "K" is a description of K in the
language of the machine.
Now, if a predicate P is expressible, its negation is expressible,
indeed by adding the symbol for the negation ~P. So if it exists a
truth predicate V, such that PA prove V("p") <-> p, by the diaogonal
lemma on ~V, there would be a K such that PA proves K <-> ~V("K"), but
from the fact that PA proves V("p") <-> p, you can get a
contradiction, and so truth is not definable.
Contrarily, provable *is* definable in arithmetic, as Godel
meticulously showed with his Beweisbar(x) (provable in German).
basically it is Ey[y is-a-proof-of x], or Ey[is-a-proof (y, x)], and
"is-a-proof (y x)" is easily definable (it is even recursive).
So Gödel's reasoning is not problematical or paradoxal. It is a valid
proof that all effective theories (where the proof are mechanically
checkable) are incomplete.
Note that there exists complete theories, but then such theory are not
effective.
Gödel's theorem is only a first theorem in a row of many
incompleteness results. A key theorem is Gödel's theorem.
I will write a longer comment in my second post of the week, when
commenting on a post by Alex. I will explain that Gödel's theorem is
quite a lucky event for mechanism in cognitive science (not in
physics!). It is capable of "saving the soul" of the machine from all
effective reductionist conceptions (a result got by Judson Webb too).
A machine (of a certain type) can already prove that if she is a
machine, then her soul is not a machine, and this by using a
translation in arithmetic of the classical definition of the soul/
knower/first person suggested by Theaetetus (Plato). That is my oldest
result, which I have extended to quantum mechanics. Quantum mechanics
is also a big chance for classical mechanism in cognitive science.
Bruno
>
> 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.
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