Date: Oct 24, 2013 9:19 PM Author: fom Subject: Re: The Invalidity of Godel's Incompleteness Work. On 10/24/2013 1:45 AM, Nam Nguyen wrote:

<snip>

>>

>> Contrary to your beliefs and Mr. Greene's misrepresentations

>> first-order logic with identity is not reducible to a mere syntactic

>> language.

>

> First-order logic with identity is reducible to the following game of

> symbol manipulation: a wff of the form x=x is an axiom of _any T_ hence

> is always provable.

>

>>

>> If that is what you wish to impose, then you are not working in

>> first-order logic with identity.

>

> I'm sorry, the above wasn't invented by Nguyen or Greene: it came from

> the founders of FOL(=) reasoning framework!

You will need to be more clear concerning which founders

you mean.

It cannot be Padoa. For he begins his paper "Logical introduction

to any deductive theory with the statement:

"If x and y are individuals [1], then x=y or ~(x-y). These are

the only relations that we can consider between individuals

without transgressing the boundaries that separate general

logic from particular deductive theories."

"[1] Whatever x and y may be, they are individuals of the

class '[equal to x] or [equal to y]'"

Notice from the footnote that what makes 'x=x' an axiom is

its extensional interpretation with respect to the class {x}.

And, the original statement clearly characterizes the symbol

'=' as one to be interpreted as a relation.

It cannot be Tarksi. Although the expression 'x=x' would

be an atomic formula in any formalized language in which

he would consider, he expressly rejects the interpretation

which you and Mr. Greene attempt to attach. Indeed, in

"The concept of truth in formalized languages" Tarski explicitly

rejects these views:

"It remains perhaps to add that we are not interested here

in 'formal' languages and sciences in one special sense of

the word 'formal', namely sciences to the signs and expressions

of which no meaning is attached."

Perhaps you refer to Carnap. He had been slow to accept

semantics in the sense of Tarski. But, even in his own

works involving purely syntactic conceptions, he distinguished

between uses of 'x=x'. Before discussing the axioms of

identity in "The Logical Syntax of Language" he has a relevant

discussion of syntactic equality:

"The two symbols 'a' and 'a' occur at different places

on this page. They are therefore different symbols (not

the same symbol); but they are equal (not unequal). The

syntactic rules of a language must not only determine what

things are to be used as symbols, but also under what circumstances

these symbols are to be regarded as syntactically equal."

Carnap's construction is too complex to present here. However,

his first use of the symbol of equality as a symbol of identity

in a logical language is restricted to numerical expressions.

In explaining its use, he writes:

"The symbol of identity or equality '=' between numerical

expressions is here intended (as in arithmetic) in the sense

that ( z_1 = z_2 ) is true, if and only if z_1 and z_2

designate the same number, to use a common phrase."

His secondary language is a typed language. Hence, identity

in this secondary language reduces to the use of equality

in arithmetic.

Carnap's later work does address semantics. In "Meaning and

Necessity: A Study in Semantics and Modal Logic" he has the

following "rule of truth",

"If A_i is an individual expression in S_i for the individual

x and A_j for y, then A_i = A_j is true if and only if x is

the same individual as y."

Notice that Carnap does not say "A_i = A_j is true if and only

if x is syntactically equal to y".

In his book "Introduction to Semantics and Formalization of

Logic" he does address a sharper distinction of some use

to you. He distinguishes between both descriptive semantics

and pure semantics as well as descriptive syntax and pure

syntax.

"Semantical investigations are of two different kinds; we

shall distinguish them as descriptive and pure semantics.

By descriptive semantics we mean the description and analysis

of semantical features either of some particularly historically

given language, e.g. French, or of all historically given

languages in general. The first would be special descriptive

semantics; the second, general descriptive semantics. Thus,

descriptive semantics describes facts; it is an empirical

science. On the other hand, we may set up a system of semantical

rules, whether in close connection with a historically given

language or freely invented; we call this a semantical system.

The construction and analysis of semantical systems is called

pure semantics. The rules of a semantical system S constitute,

as we shall see, nothing else than a definition of certain

semantical concepts with respect to S, e.g., 'designation in S'

or 'true in S'. Pure semantics consists of definitions of this

kind and their consequences; therefore, in contradistinction

to descriptive semantics, it is entirely analytic and without

factual content.

"We make an analogous distinction between descriptive and pure

syntax and divide these fields into two parts, special and

general syntax. Descriptive syntax is an empirical investigation

into the syntactical features of given languages. Pure syntax

deals with syntactical systems. A syntactical system (or calculus)

K consists of rules which define syntactical concepts, e.g.

'sentence in K', 'provable in K', 'derivable in K'. Pure syntax

contains the analytical sentences of the metalanguage which

follow from these definitions."

Now, I hope you read those passages closely. Isn't all of

that stuff about pure syntax exactly what YOU MEAN? Isn't

all of that stuff about pure syntax exactly what Mr. Greene

spouts endlessly? ... the nonsense in his remarks which makes

you think he knows anything?

WELCOME TO HERBRAND SEMANTICS!!!

"Herbrand logic differs from first-order logic solely in the

structures it considers to be models. The semantics of a given

set of sentences is defined to be the set of Herbrand models

that satisfy it, for a given vocabulary."

M |= s=t if and only if s and t are syntactically identical.

http://www.cs.uic.edu/~hinrichs/herbrand/html/herbrandlogic.html

Mr. Greene is an idiot. I have caught him in these misrepresentations

again and again and again. I am so sick and tired of it that I

killfiled him almost immediately upon his return.

Herbrand logic is *not* first-order logic with identity.

If you plan on lecturing others concerning what the founders of

first-order logic with identity had in mind, then you ought

to have taken the time to find out. Since your statement could

not possibly have come from your own knowledge in these matters, I

had to surmise that you hoped that Mr. Greene's knowledge had

some basis in fact.

It does not.

>>

>> Nor do you understand *why* this is *formally* the correct

>> interpretation. Your "syntactic elimination" depends upon

>> provability. Whatever its virtues, provability is an epistemic

>> notion and not a semantic notion. But, it is the semantic notion

>> which defines the first-order paradigm. Without an understanding

>> of the fact that the four necessary relations may not be

>> made unnecessary you have no notion of "formal" as it applies

>> to first-order logic with identity.

>

> The game of symbol manipulation is there to stay with FOL=, nonetheless.

>

If you think it a game, so be it. But, the very notion of a

deductive calculus is one of syntactic transformation rules.

No one who has tried to help you have made any representations

otherwise.

> It doesn't matter what philosophical motivation you might have had, it's

> part of the definition of reasoning with rules of inference in FOL with

> identity: either you'd conform to it, or betray it.

>

> Godel betrayed it, and so have we.

>

I have not.

And, I do not know what Goedel thought of semantics generally.

Considering his work in set theory, I suspect he understood the

matters involved.

But, the incompleteness theorem is a metamathematical theorem

directed at Hilbert's program of metamathematics. It is distinct

from the considerations which define first-order logic with

identity.

As for you, it is a different matter. You prance around here

talking about first-order logic with identity when, in fact, you

are confused by matters. You wish to restrict to purely syntactical

notions without grasping that that changes the logical paradigm.

Perhaps Peter is right. You are ineducable.