Date: Jan 29, 2013 4:41 AM
Author: fom
Subject: Re: Formally Unknowability, or absolute Undecidability, of   certainarithmeticformulas.

On 1/28/2013 11:28 PM, Nam Nguyen wrote:
> On 28/01/2013 6:20 AM, Frederick Williams wrote:
>> Nam Nguyen wrote:
>>

>>> I meant, what would "tomorrow", "today" have anything to to with
>>> _mathematical logic_ ?

>>
>> Oh, a lot. Look up 'temporal logic'. In my day it was something of a
>> curiosity of interest only to philosophers (hiss, boo, etc) but now it
>> is of much interest to computer scientists among others.

>
> It seems you aren't aware, but the assumed logic of this thread here
> is the familiar FOL=.
>


How can that be if you are requesting alternative
interpretations of quantification?

However, the answer to your question concerning "tomorrow" and
"today" is found in the relationship of model theory to
description theory.

Originally, Frege spoke of incomplete symbols such
as

x+2=5

because they require a "name" to have a "truth value".

Modern model theory is a bit senseless because they
use a parameterized theory (set theory) to justify
speaking of "truth" for an object language. If you
actually read Tarski's paper, it explicitly excludes
consideration of how the "objects" of an interpretation
transform incomplete symbols to complete symbols (those
with a truth value). This reflects the Russellian
position that "naming" is an extra-logical function.

One gets to an explicit discussion of names and indentity
within a model in Abraham Robinson's "On Constrained
Denotation". Whether or not one agrees with Robinson, it
returns the question of truth valuation to the role of
descriptions and reference.

Having gone this far, the next issue is the relation between
demonstratives and descriptions. This involves indexicals.
Kaplan produced a decent intensional logic of demonstratives
that makes plain the relation between demonstratives and
descriptions. Since it utilizes indexicals, temporal
modal operators play a role.

To say that

x+2=5

is true because

there exists an "object" y such that

y+2=5

is different from saying that

3+2=5

is true.

That is the difference between using a "set"
and a "name".

The history of description theory explains why this
is not taught in mathematical logic. But that historical
basis has been collapsing for over 50 years. This change
has simply been ignored by the mathematical community.