Re: Non standard models of PA

On Nov 1, 9:06 pm, george <gree...@xxxxxxxxxx> wrote:
On Nov 1, 9:45 am, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

Yes, in NAFL truth and falsity are indeed model-dependent. But the
NAFL model is a different animal altogether from the classical model.

"Animal" is a bit extreme. If you expect to get taken
seriously then you are going to have to concede that this is all
abstract.


To be precise, it is all in the human mind as far as NAFL is
concerned.


Here is a brief explanation of the NAFL truth definition (what I call
the "Main Postulate" of NAFL).

There are no truths in just the language of a NAFL theory T.

This is not quite consistent as you are going to explain it below.

NAFL truths are always with respect to axiomatic theories.

Exactly, so the question must arise, are THOSE theories, the ones
with respect to which we DO have truths, NAFL theories?
You have already insisted that they be axiomatic theories.
Are they axiomatic *NAFL* theories? IF SO, then this CONTRADICTS
Your top line, that "there are not truths in just the language of a
NAFL
theory T". Essentially, you are forcing truth to be related to
theory;
there is no such thing as a truth because every truth must instead be
a
T-truth, for some theory T. But you are also saying that T-truths are
NOT in the language of T itself.


No that is not what I meant. What I meant was that there are no
absolute truths for propositions in the language of NAFL theories,
i.e., for formal propositions (there are absolute truths for
propositions *about* NAFL theories, e.g "T is consistent" is a
proposition that is intrinsically either true or false, but this
proposition cannot be formalized in the language of a NAFL theory. For
formal propositions in the language of NAFL theories, there exist only
truths with respect to NAFL theories, they are *not* intrinsically
either true or false without any referencing theory.


The axioms of T are held in the human mind

This is just irrelevant.
It does not MATTER where you put your source of final authority and
it does not MATTER whether you call it "the mind" or something else.
EVERYbody is willing TO JUST CONCEDE you a source of final authority
BECAUSE OTHERWISE WE WOULD HAVE AN INFINITE REGRESS.
So we're going to just ignore this part.


No, this is all-important in NAFL. Because it is the human mind that
ultimately decides the T-truth value of a proposition P that is
undecidable in a NAFL theory T. If the human mind does not add any
axioms to T, thus leaving P undecided, then P is "neither true nor
false" with respect to T. You have not understood this correctly as
yet, as can be seen below.


I am basically saying that the theorems of
T are basically "axiomatic assertions" by the human mind, and that
there is no other truth for *formal* propositions of a theory.

No, you're basically NOT saying that.
You're basically saying that if you want to resolve truth or falsity
of
sentences in the language of a theory T, you have to go to a bigger
theory and language T* to do it.


What you have not yet grasped is that it is the human mind that
decides what T* is. T* is *not* a fixed theory. For example, let P be
a proposition that is undecidable in a NAFL theory T and let us say I
am the human being with the theory T in mind. Then I would have to
specify what T* I have in mind. I could take just about any T* that I
want, as long as T* is a legitimate NAFL theory and proves all the
axioms of T, and includes all the propositions in the language of T.
E.g I could take T*=T+P. Then P is true with respect to T according to
my interpretation. I could change this interpretation and assert
tomorrow that my T* is now T+~P. Then P is false with respect to T
according to my interpretation. Different human beings could have
different interpretations in mind at a given time. No problem if you
find P to be false with respect to T while I find it true with respect
to T. No problem if either of us changes our positions, which could be
time-dependent.


So the action now inevitably shifts to undecidable propostions.

This is not "inevitable" at all, for the sole and simple reason that
you have NOT DEFINED "undecidable". If you are going to just borrow
OUR definition then you have conceded too much, since you are
attacking
our definition as incoherent.


In NAFL undecidability is always with respect to NAFL theories. It
does not make sense in NAFL to ask if a proposition is undecidable,
period. It only makes sense to ask if it is undecidable with respect
to some theory in whose language that proposition is legitimate. When
I said "undecidable" above, some referencing theory was implicit.


Let P be a proposition (in the language of T) that is undecidable in T.

That is not generally POSSIBLE. For sufficiently rich T we don't KNOW
that
P is undecidable. The usual way we PROVE P undecidable is by
ascending
TO AN INFINITARY meta-theory, which you DISallow. IN THAT theory, we
can
construct models of both T+P and T+~P. But since you are explicitly
allowing
both infinitary reasoning AND our model theory, you don't even have a
coherent
notion of what it means for P to be undecidable.


A proposition P is undecidable in a consistent NAFL theory T if
neither P nor ~P is provable in T. Here it is taken for granted that
every proposition in the language of a NAFL theory T is intrinsically
either provable or refutable or undecidable in that theory. The notion
of provability is not formalizable in NAFL theories; is is a
proposition *about* NAFL theories. The provability or otherwise of a
propostion P in a NAFL theory T is an absolute (Platonic) truth. In
other words it makes sense to hypothesize that P is undecidable in a
consistent NAFL theory T; this is not a T-truth or a truth with
respect to any other NAFL theory; it is an absolute truth.


In NAFL, there is a model-dependent truth for these propositions that is
generated as follows.

It isn't REALLY model-dependent. As YOU define model, it may be
model-
dependent, but when the smoke clears, you are just going to have
another
NAFL THEORY deciding this. It will not be T, it will be bigger than
T.
But why don't you just ignore the stuff about the mind now and say
something
about how we get from T to T*, and how the language of T relates to
the
language of T*?


I have already explained this above. T*, like T is a NAFL theory and
resides in the human mind. Further, the human mind fixes what T* is
and its language must contain all the propostions in the language of T
and T* must at least prove all the axioms of T. I repeat, T* is not a
fixed theory, either in time or with respect to different human minds,
and this is where NAFL model differs radically from its FOL
counterpart.


Thus for every NAFL theory T held in the human mind, there is an
"interpretation" T*, which is also a NAFL theory obtained by possibly
adding axioms to T.

Well, is there ANY OTHER way to get from T to T*?
This is IMPORTANT. You HAVE to clarify this.

> Note that the NAFL model is a temporary animal that lives

in the human mind and will get changed when T* changes.

Neither T nor T* *can* change.
They are just abstract collections of sentences.


No. T* could be time-dependent. To be precise, a human mind could
change T* according to its free will. For a given T, the T-truths that
arise out of provability in T will not change. But for a T-undecidable
proposition P, the T-truths could change in time and with different
human beings at a given time; because, as I explained, T* could either
change in time for a given human mind or change with respect to
different human minds, even at one given time.


Further this
model is not "pre-existing" as in classical logic; the only warrant
for the existence of this model is that the human mind has "generated"
it via the interpretation T*.

But YOU have already insisted that T* is, LIKE T, a NAFL theory.

If we take T*=T (i.e., if the human mind has not added any axioms to
T), then a proposition P that is undecidable in T will be "neither
true nor false" in the resulting (non-classical) NAFL model of T.

Please. If T*=T and P is undecidable in T *and* undecidable in T*,
then you might as well just say that T *is* the NAFL model of T.

Yes, if a human mind decides not to add any axioms to T then for that
human mind T*=T. Then P is "neither true nor false" with respect to T.
However one has to be careful here. In the non-classical model for T
that is generated by this T* (=T), P&~P is the case, i.e. P is "true"
and ~P is "true". However these are non-classical "truths". In this
non-classical model, 'P" expresses that "~P is not provable in T*
(=T)" and "~P" expresses that "P is not provable in T* (=T)". What I
mean by "neither true nor false" is that neither P nor ~P is provable
in T*. Here P can be classically "true" ("false") with respect to T
iff it is provable (refutable) in T*. So P is *classically* "neither
true nor false" with respect to T in this sense, and that is precisely
what "P&~P" expresses in the non-classical model.

Regards, RS


.

Relevant Pages

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