Transforming formal proof into sound deduction (greatly simplified)

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PeteOlcott
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Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Mon Apr 22, 2019 6:47 am

Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) unlike the formal proofs of symbolic logic provability cannot diverge from truth.

When we consider sound deductive inference to the negation of a conclusion we now also have a definitive specification of falsity.

Within the sound deductive inference model we can be certain that valid inference from true premises derives a true conclusion.

∴ Within the sound deductive inference model any logic sentence that does not evaluate to True or False is unsound, there is no undecidability or incompleteness in this model.

The key criterion measure that the sound deductive inference model would add to the formal proofs to theorem consequences of symbolic logic would be the semantic notion of soundness.
Last edited by PeteOlcott on Mon Apr 22, 2019 10:23 pm, edited 1 time in total.

Impenitent
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Re: Transforming formal proof into sound deduction (greatly simplified)

Post by Impenitent » Mon Apr 22, 2019 4:02 pm

defining the future does not ensure its conformity to your definition...

-Imp

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Tue Apr 23, 2019 4:56 am

Impenitent wrote:
Mon Apr 22, 2019 4:02 pm
defining the future does not ensure its conformity to your definition...

-Imp
You are failing to understand the Analytic/Synthetic distinction:
https://plato.stanford.edu/entries/analytic-synthetic/

Logik
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Re: Transforming formal proof into sound deduction (greatly simplified)

Post by Logik » Tue Apr 23, 2019 11:17 am

PeteOlcott wrote:
Tue Apr 23, 2019 4:56 am
You are failing to understand the Analytic/Synthetic distinction:
https://plato.stanford.edu/entries/analytic-synthetic/
And you are failing to understand that not everybody subscribes to Kantian philosophy. Unless they choose to.

PeteOlcott
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Joined: Mon Jul 25, 2016 6:55 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Tue Apr 23, 2019 2:56 pm

Logik wrote:
Tue Apr 23, 2019 11:17 am
PeteOlcott wrote:
Tue Apr 23, 2019 4:56 am
You are failing to understand the Analytic/Synthetic distinction:
https://plato.stanford.edu/entries/analytic-synthetic/
And you are failing to understand that not everybody subscribes to Kantian philosophy. Unless they choose to.
Truth is the way that it is, one either gets this correctly or fails to. Mathematical specifications such as mine are eternal.

Logik
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Joined: Tue Dec 04, 2018 12:48 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by Logik » Tue Apr 23, 2019 3:08 pm

PeteOlcott wrote:
Tue Apr 23, 2019 2:56 pm
Truth is the way that it is, one either gets this correctly or fails to.
That is a rather interesting decision problem unfolding right before you.

We have two hypothesis on the table:

A. Pete Olcott correctly gets the way truth is.
B. Pete Olcott fails to get the way truth is.

In what formal system would you decide whether A ⇔ True, B ⇔ False; OR A ⇔ False, B ⇔ True

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Tue Apr 23, 2019 4:00 pm

Logik wrote:
Tue Apr 23, 2019 3:08 pm
PeteOlcott wrote:
Tue Apr 23, 2019 2:56 pm
Truth is the way that it is, one either gets this correctly or fails to.
That is a rather interesting decision problem unfolding right before you.

We have two hypothesis on the table:

A. Pete Olcott correctly gets the way truth is.
B. Pete Olcott fails to get the way truth is.

In what formal system would you decide whether A ⇔ True, B ⇔ False; OR A ⇔ False, B ⇔ True
Truth is what it is no matter who thinks otherwise.
Wrongheaded opinions have no effect on the actual nature of truth itself what-so-ever.

The only reliable formal system to determine what is true and what is false
with 100% perfectly justified complete certainty is the sound deductive inference model.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by Logik » Tue Apr 23, 2019 4:05 pm

PeteOlcott wrote:
Tue Apr 23, 2019 4:00 pm
Logik wrote:
Tue Apr 23, 2019 3:08 pm
PeteOlcott wrote:
Tue Apr 23, 2019 2:56 pm
Truth is the way that it is, one either gets this correctly or fails to.
That is a rather interesting decision problem unfolding right before you.

We have two hypothesis on the table:

A. Pete Olcott correctly gets the way truth is.
B. Pete Olcott fails to get the way truth is.

In what formal system would you decide whether A ⇔ True, B ⇔ False; OR A ⇔ False, B ⇔ True
Truth is what it is no matter who thinks otherwise.
Wrongheaded opinions have no effect on the actual nature of truth itself what-so-ever.

The only reliable formal system to determine what is true and what is false
with 100% perfectly justified complete certainty is the sound deductive inference model.
Well well, we have ourselves an idealist.

Great.

So you will help us get to the "100% justified completely certain truth"

Either
"Pete Olcott correctly gets the way truth is" ⇔ True
OR
Pete Olcott fails to get the way truth is." ⇔ True

And since you absolutely INSIST that: ¬(True ⇔ False) then I guess one of the options above is 100% true and correct; and the other is 100% false and incorrect.

Please help us decide.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Tue Apr 23, 2019 4:41 pm

Logik wrote:
Tue Apr 23, 2019 4:05 pm
PeteOlcott wrote:
Tue Apr 23, 2019 4:00 pm
Logik wrote:
Tue Apr 23, 2019 3:08 pm

That is a rather interesting decision problem unfolding right before you.

We have two hypothesis on the table:

A. Pete Olcott correctly gets the way truth is.
B. Pete Olcott fails to get the way truth is.

In what formal system would you decide whether A ⇔ True, B ⇔ False; OR A ⇔ False, B ⇔ True
Truth is what it is no matter who thinks otherwise.
Wrongheaded opinions have no effect on the actual nature of truth itself what-so-ever.

The only reliable formal system to determine what is true and what is false
with 100% perfectly justified complete certainty is the sound deductive inference model.
Well well, we have ourselves an idealist.

Great.

So you will help us get to the "100% justified completely certain truth"

Either
"Pete Olcott correctly gets the way truth is" ⇔ True
OR
Pete Olcott fails to get the way truth is." ⇔ True

And since you absolutely INSIST that: ¬(True ⇔ False) then I guess one of the options above is 100% true and correct; and the other is 100% false and incorrect.

Please help us decide.
https://en.wikipedia.org/wiki/Stipulative_definition
A stipulative definition is a type of definition in which a new or
currently-existing term is given a new specific meaning for the
purposes of argument or discussion in a given context.

I am stipulating:
T ⇔ True
⊥ ⇔ False
I am also stipulating that all of the definitions and relations of predicate logic.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by Logik » Tue Apr 23, 2019 4:43 pm

PeteOlcott wrote:
Tue Apr 23, 2019 4:41 pm
https://en.wikipedia.org/wiki/Stipulative_definition
A stipulative definition is a type of definition in which a new or
currently-existing term is given a new specific meaning for the
purposes of argument or discussion in a given context.

I am stipulating:
T ⇔ True
⊥ ⇔ False
I am also stipulating that all of the definitions and relations of predicate logic.
Aristotle did that 2000 years ago.

Don't you feel like you are stealing Stanford's money after 22 years ?

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Tue Apr 23, 2019 4:58 pm

Logik wrote:
Tue Apr 23, 2019 4:43 pm
PeteOlcott wrote:
Tue Apr 23, 2019 4:41 pm
https://en.wikipedia.org/wiki/Stipulative_definition
A stipulative definition is a type of definition in which a new or
currently-existing term is given a new specific meaning for the
purposes of argument or discussion in a given context.

I am stipulating:
T ⇔ True
⊥ ⇔ False
I am also stipulating that all of the definitions and relations of predicate logic.
Aristotle did that 2000 years ago.

Don't you feel like you are stealing Stanford's money after 22 years ?
I closed the expressiveness gap of formal proofs to theorem consequences of symbolic logic
by converting these formal proofs to conform to the sound deductive inference model.

Logik
Posts: 4041
Joined: Tue Dec 04, 2018 12:48 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by Logik » Tue Apr 23, 2019 5:06 pm

PeteOlcott wrote:
Tue Apr 23, 2019 4:58 pm
I closed the expressiveness gap of formal proofs to theorem consequences of symbolic logic
by converting these formal proofs to conform to the sound deductive inference model.
In my personal experience rules hinder expressiveness, but lets skip that for a second.

In what way is the deductive inference model different to proof theory?

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Tue Apr 23, 2019 5:12 pm

Logik wrote:
Tue Apr 23, 2019 5:06 pm
PeteOlcott wrote:
Tue Apr 23, 2019 4:58 pm
I closed the expressiveness gap of formal proofs to theorem consequences of symbolic logic
by converting these formal proofs to conform to the sound deductive inference model.
In my personal experience rules hinder expressiveness, but lets skip that for a second.

In what way is the deductive inference model different to proof theory?
Undecidability, and incompleteness cannot exist in the former.
When redefining formal proofs to conform to sound deduction the incompleteness theorem is refuted.

wtf
Posts: 822
Joined: Tue Sep 08, 2015 11:36 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by wtf » Wed Apr 24, 2019 2:20 am

PeteOlcott wrote:
Mon Apr 22, 2019 6:47 am
Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) unlike the formal proofs of symbolic logic provability cannot diverge from truth.
Aren't there mutually exclusive but independently self-consistent axiomatic systems?

For example the axioms of geometry with the parallel postulate, and the axioms of geometry without the parallel postulate.

I still don't understand how you deal with that situation.

In my understanding of the terms, a deduction in Euclidean or non-Euclidean geometry can be valid, meaning it goes from premises to conclusion by a legal sequence of applications of the inference rules.

But neither system can be said by logicians or geometers to be true. That question is a matter for the physicists.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

Re: Transforming formal proof into sound deduction (greatly simplified)

Post by PeteOlcott » Wed Apr 24, 2019 3:09 am

wtf wrote:
Wed Apr 24, 2019 2:20 am
PeteOlcott wrote:
Mon Apr 22, 2019 6:47 am
Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) unlike the formal proofs of symbolic logic provability cannot diverge from truth.
Aren't there mutually exclusive but independently self-consistent axiomatic systems?

For example the axioms of geometry with the parallel postulate, and the axioms of geometry without the parallel postulate.

I still don't understand how you deal with that situation.

In my understanding of the terms, a deduction in Euclidean or non-Euclidean geometry can be valid, meaning it goes from premises to conclusion by a legal sequence of applications of the inference rules.

But neither system can be said by logicians or geometers to be true. That question is a matter for the physicists.
https://www.cs.unm.edu/~joel/NonEuclid/ ... idean.html
That took me almost five minutes to figure out.
Most of the time was finding out the details of Non-Euclidean Geometry.

My system already handles that by defining truth to be relative to a formal system:
Axiom(1) ∀F ∈ Formal_System ∀x ∈ Closed_WFF(F) (True(F, x) ↔ (F ⊢ x))
Axiom(2) ∀F ∈ Formal_System ∀x ∈ Closed_WFF(F) (False(F, x) ↔ (F ⊢ ¬x))

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