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 andits relevance to Gödel’s Theorem in the hope of putting an end todiscussion of a topic somewhat tangential to the main one. Beforedoing so, I am forwarding an email from Lou Kauffman which gives avery 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 forumcomments for this week.I am going to discuss a Smullyan puzzle in detail with you. I call this the Smullyan Machine. THE SMULLYAN MACHINEThe machine has a button on the top and when you press that button,it prints a string of symbols using the following three letteralphabet.{ 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 (givenmeaning) as follows:Meaningful Strings(When I say “X can be printed by the Machine” I mean that when youpress 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 ~PPRThis has meaning and it states that the machine cannot bring PR allby itself when the button is pressed.AXIOM OF THE MACHINEThe Smullyan Machine always tells the truth when it prints ameaningful string.THEOREM. There is a meaningful string that is true but not printableby the Smullyan Machine.PROOF. Let S = ~PR~PR. This string is meaningful since it startswith ~PR.Note that S = ~PRX where X = ~PR. Thus by the definition (above) ofthe meaning of S, “XX is not printable by the Machine.”We note however that XX = ~PR~PR = S. Thus S has the meaning that “Sis not printable by the Machine.”Since the Machine always tells the truth, it would be in acontradiction 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 truebut not printable string. The completes the proof.————————————————————————————————————————————————————— Now I have an assignment for you.Please criticize the Smullyan Machine from your phenomenologicalpoint of view.If you wish you could include my description of the Machine and makea statement about it on FIS.My point and Smullyan’s point in his Oxford University Press Book onGodel’s Theorem, is that the Machine is an accurate depiction of theGodel argument, withPrintabilty replacing Provablity. The way that self-reference workshere, and the way the semantics and syntax are controlled is verymuch like the way these things happen in thefull Godel theorem. The Machine provides a microcosm for thediscussion 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 distinguishsyntactic self-reference which is reference to the words or sentencethat one is using, from semantic self-reference, which is referenceto the MEANING of the words or sentences one is using. There isnothing 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 thatstatement is not printable. On the supposition that such is thecase, the statement it makes must also be semantically self-referential for the following reason. In Smullyan’s scenario, theprinting machine prints only true statements. As a result, asentence 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 nottrue’ 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 isconsequently logically equivalent (each entails the other) to astatement that says of itself that it is not true, that is, it isequivalent to a Liar statement.As such, it is semantically incomplete or vacuous; it does not makea statement, and hence is neither true nor false, and so cannotpossibly 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’ hasbeen incorrectly interpreted. The second expression ‘~PR’ at the endof Smullyan’s sentence is a well-formed formula in Smullyan’ssystem, but when translated into English, it has no grammaticalsubject, and so cannot be a sentence; it is merely a predicate, andso does not make a statement. Hence Smullyan’s sentence must besaying that the string of symbols, ‘~PR’, translatable as thepredicate ‘is not printable’, is not printable.On this second interpretation of the Smullyan sentence, ‘~PR~PR’ isstill a sentence that cannot be printed by a machine that printsonly 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 containsthat string of words, it cannot be printed. On the other hand, ifwhat the sentence says is false, it cannot be printed because theprinter prints only what is true. The Smullyan sentence, whether thestatement it makes is true or a false, cannot be printed by aprinter that prints only sentences that make true statements. Itcould, of course, be printed by a different printer, one that alsoprints false statements such as it.On this second interpretation of ‘~PR~PR’, the Gödel sentencediffers from Smullyan’s sentence in that its subject is a sentence,not a predicate. It states that a certain sentence, itself, is notprovable in a certain formal system. The sentence allegedly makes anarithmetical statement on its intended interpretation, but since itis semantically self-referential (like the statement that thisstatement is true) it is vacuous and so says nothing, much lesssomething that is true. Because it says nothing, it has no businessbeing 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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