Re: The fallacy of strengthened liar's paradox.
- From: "R. Srinivasan" <sradhakr@xxxxxxxxxx>
- Date: Tue, 1 Jan 2008 08:26:51 -0800 (PST)
On Dec 26 2007, 4:21 am, Newberry <newberr...@xxxxxxxxx> wrote:
On Dec 25, 2:47 pm, John Jones <jonescard...@xxxxxxx> wrote:That is not obvious at all. I agree with John Jones. To see what the
On Dec 24, 11:11�pm, Newberry <newberr...@xxxxxxxxx> wrote:
Let P be the sentence "This sentence is meaningless." Is it true or
false? It should not be difficult to answer. Tractatus Logico-
Philosophicus says: "In order to tell whether a picture is true or
false we compare it with reality." [2.223] When we attempt to compare
"This sentence is meaningless" with reality we find that it is not
comparable with anything. It is not a picture of a fact; it is
meaningless.
We can analyze the situation further:
Case A: P is true.
If P is true. Then it is the case that it is meaningless. But then it
cannot be true. This is a contradiction. Therefore P is not true.
Case B: P is false.
If P is false then it is not the case that it is meaningless. It is
the opposite of what it claims. This is a contradiction. Therefore �P
is not false.
Case C: P is meaningless.
If P is meaningless then nothing is the case. There is no
contradiction. In a three valued logic (T, F, M) we conclude that P is
meaningless.
It is not correct to say that if the sentence is meaningless than what
it SAYS is true. This argument assumes that it becomes TRUE half way
through the argument and then it IS THE CASE that it is meaningless.
Thus the sentence confirms our initial assumption that it was
meaningless. But if it stays meaningless all the time it confirms
nothing.
Clearly, if the sentence does not have any meaning then it does not
have the meaning that it is meaningless.
This gives us the basic insight that all self-referential, paradoxical
sentences, including possibly Goedel's sentence, are probably
meaningless.
Let P be the sentence "This sentence is meaningless." Is it true or
false?
I can't start this car. You have not said what sort of object P a
sentence is, nor even specified what sentence you refer to by 'it'.
"This" in the sentence "This sentence is meaningless" obviously refers
to the sentence in quotation marks.
problem is, consider the following thought experiment:
Lat us say Mr. X blurts out "This sentence is....." and exactly at
that point he drops dead. Can you now let us know what is the sentence
that Mr. X had in mind? The answer is we cannot be sure at all. All we
can say is that if Mr. X had no sentence in mind at the time of his
death, he was attempting to attribute some property to a non-existent
sentence. If the sentence itself is non-existent, then any attributed
property to it (e.g. "meaninglessness") is also non-existent.
The problem here is that we human beings can parse sentences only when
their words are read sequentially, which means that the informal
notion of "time" is important in understanding what sentences mean.
Keeoing this in mind, let us re-formulate the sentence in question in
an *exactly* equivalent manner as follows:
"This sentence, which has not yet been defined, is meaningless"
Now we can clearly agree that the above sentence attempts to attribute
the property of "meaninglessness" to a non-existent sentence. Which
makes no sense whatsoever. Yet what is wrong with the above
formulation? If a human being who utters the above sentence drops dead
after uttering "defined", (s)he has clearly not yet defined any
sentence whatsoever, even under the assumption that "This" in the
above sentence points to the same sentence being constructed.
Conclusion: When somebody utters "This sentence is blah blah blah..."
then s(he) should have a *constructed* sentence in mind the moment
"sentence" is uttered. Then and only then is it legal to attribute
properties like "meaningful" or "meaningless" to that sentence.
In order to make sense of "This sentence is meaningless", the sentence
referred to must be taken as having a Platonic esistence, independent
of our self-referential attempt at "definition". The problem is that
Godel's reasoning enables us to formalize such Platonically existing
objects (e.g. "sentence", "theory") as formal mathematical objects of
some universe. This leads to paradox.
In NAFL also sentences are taken to be pre-existing, when formulating
theories. Theories are also pre-existing objects. E.g. "Every
proposition in the language of a NAFL theory is either provable of
refutable or undecdiable in that theory" is an absolute (Platonic)
truth in NAFL, as it is in classical logic. But the important
distinction is that NAFL does not accept that the notion of "sentence"
or "theory" itself can be formalized as an object of the universe.
Sentences and theories are used to make assertions *about* objects of
the universe and cannot themselves be objects of the same universe.
They are essentially "meta-mathematical" objects whose existence
cannot be formalized within NAFL theories. The way this conclusion
comes about is via the axiomatic nature of NAFL truth for formal
sentences in the languages of NAFL theories, as opposed to the
Platonic nature of classical truth.
For the NAFL truth definition,see the recent sci.logic thread "FOL/
Intuitionistic logic versus NAFL. Part 1. Failure of non-
contradiction" at the following link:
http://groups.google.co.bw/group/sci.logic/browse_thread/thread/48894ac0f1d11787/f34781b18deff9c4?#f34781b18deff9c4
I will shortly be starting another thread (Part 2), which explains the
definition of NAFL theories.
Regards, RS
.
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