# [Fis] Gödel discussion

```Greetings everyone,
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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.
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-------- 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
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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.
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Strings that begin with P, ~P, PR or ~PR are INTERPRETED (given meaning) as follows:
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Meaningful Strings
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(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
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This has meaning and it states that the machine cannot bring PR all by itself when the button is pressed.
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AXIOM OF THE MACHINE
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The Smullyan Machine always tells the truth when it prints a meaningful string.
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THEOREM. There is a meaningful string that is true but not printable by the Smullyan Machine.
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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.
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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:

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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.
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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.
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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.
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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.
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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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