Re: Technique and Foundation

John Jones schrieb:
As it traditionally stands, as you say, a function can have any interpretation. But that is not true. All the interpretations, I think you will find, present a particular class of object behaviours.

What do you mean by an interpretation that I can find? Could you please
elaborate? It doesn't matter what I find as an exo-reference for
example for an axiom. When I am doing a proof from the axioms, the
axioms always carry any interpretation as far as the proof allows.
And this is much more that I can imagine.

This is fundamental to logic. It has been expressed in meta-logical
results by things such as the löwenheim-skolem theorem, the compactness
theorem and by the existence of non-standard models etc..

Objects can be incommensurable, and bear their own particular rules. Negation, for example, only works with a particular class of objects. So logic cannot treat every sign the same. Signs are hybrid when they bear

I didn't say that each and every symbol is the same. But if you have
negation at one place in an axiom and negation in another place in an
axiom, its the same negation. It is co-referent. It referes to the
same truth table, namely:

A | ~A
-------
0 | 1
1 | 0

The same holds for the function symbols, the relation symbols, etc..
They are coreferent what their basic interpretation concerns. It
doesn't mean that their interpretation cannot be modified by the
current context. But before the context is built, every starts with
co-reference. This is fundamental.

objects behaviours that are incommensurable. for example the Godel sentence is both a referencing and a self-referencing sign. This is a hybrid and inadmissable.

If you want to cast the different sorts of symbols that exist behind
the notion of incommensurability, then well go on. But this would be
the only application of this notion in logic. Otherwise I don't see
any use of this notion.

Also please explain which sign it is that makes the goedel sentence, referencing and self referencing. Because the godel sentence is
a composite sign, but the elementary signs in the godel sentence
are co-referent as I said. There is no doubt.

The composite sign of the godel sentence is not refererncing or
self referencing, this is your exo-referential interpretation. Such
concepts do not exist in the logic itself. That is the reader
of an axiom or formula, that does make this up. For example when
we have the following formula:

G <-> ~[]G.

Then the two occurences of the variable symbol G, namely the occurence
1 (left) and the occurence 2 (right), are co-referent, thats all.
Now if we substitute for G a real sentence, that is a composite sign,
then all the signs on the left are co-referent with the their
counterpart signs on the right. Thats all.

And the co-reference is the main reason that makes a sentence
as above unsatisfiable. Lets drop the modal operator [] for a moment.
And lets look at X <-> ~X. Because the first occurence of X and
the sencond occurence of X are correferent, they receive the same
truth value either true or false concurrently. And thus the
composite sign, will never receive a truth value true. Because:

A B | A <-> ~B X | X <-> ~X
-------------- ------------
0 0 | 0 <<<<< 0 | 0
0 1 | 1
1 0 | 1
1 1 | 0 <<<<< 1 | 0

The correference forces us, that we can only look at the first and
the last row of the truth table. Thats the virtue of having two times
an X in the formula. Without co-reference there would be no
Goedel sentence.

Don't mention axioms to me again. Axioms work with a particular set of objects that obey particular rules. I am saying that some objects have rules that cannot be incorporated in traditional axioms.

Agreed that objects might have rules that can not incorporated as exo-references in axioms. But this is again about expectations. As I mentioned in my intial post, don't expect too much from logic. Its
your fault when your own expectations are too high, and they are
not met.

But axioms don't work with particular objects that obey particular
rules. Axioms have symbols in it, that refer to any kind of objects.
And the same symbol occuring in different places in the axiom, refers
to the same object at one momemt of interpretation. Except maybe for
bound variables, where we have to be a little bit more careful.

Tandem functionality is more a statement supporting the idea of a consistent all-embracing system, because there is no such thing as the 'same sign' occuring in 'another place'. Equals for equals demands a universe whose objects are all alike.

Maybe in natural language there is no such thing as the same sign occuring in another place denoting the same thing. But its
fundamental for logic. And you are seeking out the fundamentals
of logic. So lets face them.
.

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