Re: A question about FOL theories and models

Hi

If the theory has a model, then proving
X implies that ~X does not follow.

But you might also not be able to
proof either X or ~X, even if the
theory has a model.

He He

Bye

Russell Easterly wrote:

"Nam Nguyen" <namducnguyen@xxxxxxx> wrote in message news:w6bHg.448896$iF6.31548@xxxxxxxxxxx


MoeBlee wrote:


If I ask someone if the number he's holding on his hand is 10, I'd
expect either a "yes", or "no" - but not something hypothetical like
"if it's a 10 then the answer is 'yes'"! While formulae of the form
(P => Q) are often satisfactory as an answer on the 1st order level, on
the meta level, hypothetical statements are often indicative of limit
in our reasoning knowledge, and less satisfactory. E.g., If I very much
would like to know if ZFC is consistent, the answer "*if* ZFC has a
model then it is consistent" wouldn't at all be satisfactory imo.


the relative consistency
proofs, the meta theorem that if ~GC is true then it's provable in PA,
... to say a few. What I'm saying (or proposing) instead of
"sporadic" acknowledgement, let's review FOL as a whole and bring
some sort of relativity principle to the very foundation of FOl. In
that way it seems to me that we would have a more coherent acknowledgement.


What kind of relativity principle?

I certainly don't have all the guidelines or details for all the
principles that I think could be used in the new foundation.
I'm still at it, but I think I could list some here:

(1) guideline 1: the limitation of knowledge has to be acknowledged,
and formalized. One specific principle under this umbrella would be
the so-called "gAI" principle that I've occasionally mentioned.

(2) guideline 2: at any level n of mathematical reasoning, there exists
a level n+1 from which we can introspect the reasoning in lower
level. The lowest level would be the one where we do FOL inferences
and reasoning. (We however accept that we can not perform the
reasoning introspection in infinite number of levels). A specific
(but incomplete) example of how this guideline could be formalized
would be that a 1st order formula could actually range over a
collection of syntactically isomorphic theories.

Again, I don't have all the specifics for now. I think (1) could be
labelled as "passive relativity" principle: mathematical knowledge
would be subjected (thus relative) to the individual's endowed ability
to know. Given any 2 individual reasoners, they might possess some
common knowledge; but one may possess some other knowledge the other
doesn't.

(2) would be what I think of as "interpretation relativity" principle:
formal semantic and truth-values could vary from one level of
reasoning to the other.


As one of the cranks, I can tell you from experience, it is very difficult
to prove if a system that allows proof by contradiction is consistent.
I don't think proving X is really enough to let us assume we can't prove ~X.

There has to be some way to limit where negation can be used.
Negation is well defined in binary logics, but it can be defined in many
ways using multi-valued logics like a 3-valued logic.

What does negation mean if the truth values are True, False, and Undecidable?
~True = False
~False = True
~Undecidable = Undecidable?

What if the truth values are 0, 1, and 2?

~0 = 1
~1 = 2
~2 = 0?

~0 = 1 or 2
~1 = 0 or 2
~2 = 0 or 1?

I usually use this last definition for negation which becomes:
~True = False or Undecidable
~False = True or Undecidable
~Undecidable = True or False

But, the "meaning" of negation seems to depend
on the interpretation of the logic values.

I like proofs by contradiction, but I can't really justify
my confidence in them.


Russell
- 2 many 2 count



.

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