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.