OUTGOEDELING A HUMAN?

Hi, please, remember I'm LauLuna.

Recently Daryl McCullough in the thread 'Is Truth Mysterious?'
proposed the following sentence:

'LauLuna will never believe this sentence'

That sentence may not seem interesting to everyone because it could be
seen as not necessarily paradoxical. LauLuna could come to believe it
when he is drunk or when he does not remember that 'LauLuna' is a nick
for him/herself in sci-logic; then the sentence would be contingently
false.

In order to get a less disputable case, let's assume that LauLuna will
always retain some constant intellectual capabilities, so that the
sentence may be regarded as ultimately referring to those
capabilities.

Consider this labelled propositionalist version and call it 'L':

'LauLuna does not believe that L expresses a true proposition'

I think this kind of construction stems from the well known article by
Whitely 'Mind, Machines, and Gödel: A Reply to Mr. Lucas', that
Hofstadter made popular. It looks like an attempt of Gödelian
diagonalization out of a human being. Whitely tried to outgoedel Lucas
by means of a sentence like:

'Lucas cannot consistently believe this sentence'

Some people conclude that the sentence is true but Lucas will never be
able to know it. But I have never seen how Lucas could be prohibited
from following the reasoning that would lead any other to conclude
that the sentence is true! I have always thought it was paradoxical
and non propositional. Lucas would have to think: the rest of the
world knows that the sentence is true but I do not know it. Too odd
indeed.

Some people believe we know nothing about sentences like L. In my
humble opinion there are a couple of things I can say about it.
Namely:

1) I'm LauLuna.
2) If I believed L, I would make it false.
3) I don't believe L.
4) If L expressed a proposition, it would say exactly what I have said
in 3)
5) If L expressed a proposition, I would have to believe it, since I
believe what I've said in 3).
6) Since I don't believe L, L expresses no proposition.

I'm entitled to conclude whatever I have concluded above. To believe
this you must only believe that I can behave consistently enough to
conclude 3) from 1) and 2) and not believe L.

So, the attempt to outgoedel me has failed, it hasn't produced a true
LauLuna-undecidable sentence but a non propositional one, a paradox.

Let's turn to the conditions necessary for a human H not to be
outgoedelizable by means of a sentence G:

'H does not believe that G expresses a true proposition'

where 'H' unambiguously identifies H. These conditions are some common
linguistic and logical skills of H, roughly the following (I find it
extremely tedious to work this out, they are the skills I've used
above 1) through 5); you might as well skip them):

1. H is able to identify him/herself as the one named in the subject
of G.
2. H is able to interpret G so as to know what proposition G would
express if it expressed any; consequently, H knows that if he/she
believed G, G would be false.
3. H's beliefs are closed regarding G are closed under elemental logic
rules.
4. H is aware of his beliefs and disbeliefs concerning G.
5. H is consistent as regards G: H will never believe G if he/she
knows G is not true, and he/she believes G if he/she believes what G
says, if anything, because H knows what G would say if G said
anything.

Assume this list is correct. If that is so, under conditions 1.
through 5., H does not believe G but he/she believes the proposition
that G would express if it expressed any. This suffices to pronounce G
non propositional, because if G expressed a proposition, H would
correctly believe it while it is false. So, G is not only non
propositional for H but also non propositional simpliciter.

So we must conclude that, under the specified conditions, G expresses
no proposition and is paradoxical. This is perhaps what happens with L
about LauLuna.

Whether H believes G or not seems to be a well defined state of
affairs that has either to be or not to be the case; this is also the
problem with the Strengthened Liar. So, how could G not express any
proposition? It can because whether H believes L or not, is not always
a well defined state of affairs: it is not so for H while H utters or
reads G for the first time; it is phenomenologically impossible for
anyone to think: 'I do not believe what I am thinking just now'. No
intentional act can be its own intentional object.

Anyone can utter (and mentally utter) a sentence like this but it
would not really be expression of the thought the sentence
linguistically points to. Rather anyone who tries to assert such a
sentence gets involved in a vain attempt to make out what he/she is
really thinking at the same time he/she is thinking it.

Now, assume that I'm a physically realized Turing machine and that
'LauLuna' stands in L for a complete description of that machine. Then
certainly L would express a proposition, because the behavior of a
mechanical device is always a well defined physical and arithmetical
state of affairs for anyone, so that no paradox is possible about it.
The point is that Turing machines can be completely described without
reference to intentional acts.

Thus, if LauLuna were a Turing machine satisfying conditions 1.
through 5. L would be propositional and non propositional at a time.

We must conclude that there is no Turing machine able to satisfy
conditions 1. through 5. And no wonder, since those conditions imply
the capability of understanding L but not believing it, while
believing the proposition which L would express if it expressed any,
and hence the capability of distinguishing sentences (syntactical
objects) from propositions (semantical objects), so that semantics
becomes functional. But a Turing machine is functionally a pure
syntactical object which cannot have the distinction sentence/
proposition as a functional feature.

That's why in my best moments I tend to think I'm not a Turing
machine.

But some of you may find unconvincing my argument. No problem: it
could just be a sign that you are simple Turing machines ;-)

I'll post an extremely tedious formalization of this in my next
message in this thread.

Regards

.

Relevant Pages

  • Re: OUTGOEDELING A HUMAN?
    ... Whitely tried to outgoedel Lucas ... So, the attempt to outgoedel me has failed, it hasn't produced a trueLauLuna-undecidable sentence but a non propositional one, a paradox. ... 'LauLuna' stands in L for a complete description of that machine. ... ifLauLunawere a Turing machine satisfying conditions 1. ...
    (sci.logic)
  • Re: OUTGOEDELING A HUMAN?
    ... 'LauLuna will never believe this sentence' ... Whitely tried to outgoedel Lucas ... LauLuna-undecidable sentence but a non propositional one, a paradox. ... if LauLuna were a Turing machine satisfying conditions 1. ...
    (sci.logic)
  • Re: OUTGOEDELING A HUMAN?
    ... LauLuna could come to believe it ... so that he is e.g. sound. ... LauLuna-undecidable sentence but a non propositional one, a paradox. ... The point is that Turing machines can be completely described without ...
    (sci.logic)
  • Re: Intensional Operators Introduction in Hypothetical Reasoning
    ... LauLuna writes: ... But an apparently similar derivation is ... used in the proof of Fitch's theorem (or paradox, ... -- James S Harris compromises ...
    (sci.logic)
  • Re: Is Truth Mysterious?
    ... LauLuna wrote: ... propositions, it becomes clear for the first time that there are two ... versions of the paradox: the sentential and the tokenist version ...
    (sci.logic)
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