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Topic: The Invalidity of Godel's Incompleteness Work.
Replies: 87   Last Post: Oct 25, 2013 2:44 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: The Invalidity of Godel's Incompleteness Work.
Posted: Oct 24, 2013 9:19 PM

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
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

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
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.

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.

Date Subject Author
10/4/13 namducnguyen
10/5/13 Peter Percival
10/6/13 LudovicoVan
10/6/13 LudovicoVan
10/9/13 fom
10/18/13 Peter Percival
10/18/13 namducnguyen
10/19/13 Peter Percival
10/19/13 fom
10/19/13 Peter Percival
10/19/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 fom
10/19/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 fom
10/19/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 fom
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/24/13 namducnguyen
10/24/13 fom
10/24/13 namducnguyen
10/24/13 Peter Percival
10/24/13 namducnguyen
10/24/13 Peter Percival
10/24/13 fom
10/24/13 fom
10/20/13 fom
10/25/13 Rock Brentwood