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.
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