fom
Posts:
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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 >> firstorder logic with identity is not reducible to a mere syntactic >> language. > > Firstorder 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 >> firstorder 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 ~(xy). 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 firstorder 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* firstorder logic with identity.
If you plan on lecturing others concerning what the founders of firstorder 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 firstorder 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 firstorder 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 firstorder logic with identity.
As for you, it is a different matter. You prance around here talking about firstorder 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.

