Date: Jan 29, 2013 4:17 AM
Author: fom
Subject: Re: Formally Unknowability, or absolute Undecidability, of certain<br> arithmetic formulas.

On 1/28/2013 11:38 PM, Nam Nguyen wrote:
> On 28/01/2013 12:06 AM, fom wrote:
>> On 1/27/2013 11:22 AM, Nam Nguyen wrote:

>>> In this thread, we propose a solution to this differentiation
>>> difficulty: semantic _re-interpretation_ of _logical symbols_ .

>> It sounds more like "coordinated interpretation."
>> That is what mathematical realism is already doing.
>> The existence quantifier is co-interpreted with some
>> notion of truth. This is the historical debate
>> from description theory addressing presupposition failure.
>> One of the foundational insights of Frege's researches
>> was to interpret contradiction existentially. In
>> contrast, Kant interpreted contradiction modally.
>> This would suggest non-existence and impossibility
>> are already coordinated in such a way that the
>> two forms of logic branch at the outset.
>> There are, of course, intensional logics that
>> mix the senses of these logics. This is where
>> the terms "de re" and "de facto" find their
>> nuanced meanings in relation to quantifier-operator
>> order.
>> No one, of course, has tried to use anything
>> like an arithmetical numbering to provide
>> correlated, but distinct, model theories to
>> interpret a single situation (quantificational
>> logic) so as to eliminate irrelevant modal
>> possibilities.

> Would you have any link on "coordinated interpretation"?

I was simply paraphrasing what your proposal sounds like.

> I'm not sure if all of those logic's would be related to my proposal
> here, which is simply re-interpreting the logical symbols _ in any_
> _which way_ one would feel pleased, provided that:
> a) The re-interpretations be cohesively meaningful (and logical).
> b) Certain corresponding provision for formula's truth and falsehood
> be available.

Look for work on "free logics." There are axiomatizations
that define "existential import." Consequently, the usual
existential quantifier is primitive, but the model theory
supports quantification over a class partitioned into
substantive and non-substantive objects.