[Fis] Gödel discussion

[Albert A Johnstone]

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.)

  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.