Re: The philosophy of logic reformulates existing ideas on a new basis --- infallibly correct

On 11/9/2024 2:53 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 11/9/2024 1:32 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

 

The assumption that ~Provable(PA, g) does not mean ~True(PA, g)
cannot correctly be the basis for any proof because it is only
an assumption.

 

It is an assumption which swifly leads to a contradiction, therefore must
be false.

 

You just said that the current foundation of logic leads to a
contradiction. Too many negations you got confused.

 I did not say that, at least I didn't mean to.  You've trimmed the
context unusually severely, so it's difficult to see what I did say.
 

When we assume that only provable from the axioms
of PA derives True(PA, g) then (PA ⊢ g) merely means
~True(PA, g) THIS DOES NOT LEAD TO ANY CONTRADICTION.

 I can't make out your weasel word "derives".  There are true things in
any system which can't be proved in that system.  Unless that system is
inconsistent, or so restricted in scope that it can't do counting.
 

But you don't understand the concept of proof by contradiction, and
you lack the basic humility to accept what experts say, so I don't
expect this to sink in.

 

We know, by Gödel's Theorem that incompleteness does exist.  So the
initial proposition cannot hold, or it is in an inconsistent system.

 

Only on the basis of the assumption that
~Provable(PA, g) does not mean ~True(PA, g)

 

No, there is no such assumption.  There are definitions of provable and
of true, and Gödel proved that these cannot be identical.

 

*He never proved that they cannot be identical*

 This is another example of lying by lack of expertise.  You are simply
wrong, there.
 

The way that sound deductive inference is defined
to work is that they must be identical.

 Whatever "sound dedective inference" means.  If you are right, then
"sound deductive inference" is incoherent garbage.
 

*Validity and Soundness*
A deductive argument is said to be valid if and only
if it takes a form that makes it impossible for the
premises to be true and the conclusion nevertheless
to be false. Otherwise, a deductive argument is said
to be invalid.
A deductive argument is sound if and only if it is
both valid, and all of its premises are actually
true. Otherwise, a deductive argument is unsound.
https://iep.utm.edu/val-snd/
Thus your ignorance and not mine.

A conclusion IS ONLY true when applying truth
preserving operations to true premises.

 I'm not sure what that adds to the argument.
 

It is already specified that a conclusion can only be
true when truth preserving operations are applied to
expressions of language known to be true.
That Gödel's proof didn't understand that this <is>
the actual foundation of mathematical logic is his
mistake.
Unprovable in PA has always meant untrue in PA when
viewed within the deductive inference foundation of
mathematical logic.

It is very stupid of you to say that Gödel refuted that.

 It is a lie to allege I said that.  I didn't.  Gödel reached his result
precisely by following truth preserving transformations on known correct
premises.  All mathematicians do.
 

-- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

 

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer