Transforming formal proof into sound deduction (greatly simplified)

[wtf]

\
That took me almost five minutes to figure out.

Then explain it to me. In English. It's perfectly clear (and has been since 1840) that no axiomatic system can reveal -- or even express -- truth. Truth is an empirical matter.

Are you telling me that you never heard of non-Euclidean geometry before I mentioned it? And felt the need to show me a link?

How can you claim to know anything about axiomatic systems when it's clear you've never taken five minutes in your life to examine one? You fail this test on every example I give you. But non-Euclidean geometry is especially famous because of some guy named Einstein.

[PeteOlcott]

\
That took me almost five minutes to figure out.

Then explain it to me. In English. It's perfectly clear (and has been since 1840) that no axiomatic system can reveal -- or even express -- truth. Truth is an empirical matter.

Are you telling me that you never heard of non-Euclidean geometry before I mentioned it? And felt the need to show me a link?

How can you claim to know anything about axiomatic systems when it's clear you've never taken five minutes in your life to examine one? You fail this test on every example I give you. But non-Euclidean geometry is especially famous because of some guy named Einstein.

Axiom(1) ∀F1 ∈ Formal_System ∀x ∈ Closed_WFF(F1) (True(F1, x) ↔ (F1 ⊢ x))
Axiom(1) ∀F2 ∈ Formal_System ∀x ∈ Closed_WFF(F2) (True(F2, x) ↔ (F2 ⊢ x))

F1 is Euclidean geometry
F2 is non-Euclidean geometry

[wtf]

Axiom(1) ∀F1 ∈ Formal_System ∀x ∈ Closed_WFF(F1) (True(F1, x) ↔ (F1 ⊢ x))
Axiom(1) ∀F2 ∈ Formal_System ∀x ∈ Closed_WFF(F2) (True(F2, x) ↔ (F2 ⊢ x))

F1 is Euclidean geometry
F2 is non-Euclidean geometry

But which is true?

[PeteOlcott]

Axiom(1) ∀F1 ∈ Formal_System ∀x ∈ Closed_WFF(F1) (True(F1, x) ↔ (F1 ⊢ x))
Axiom(1) ∀F2 ∈ Formal_System ∀x ∈ Closed_WFF(F2) (True(F2, x) ↔ (F2 ⊢ x))

F1 is Euclidean geometry
F2 is non-Euclidean geometry

But which is true?

That was what was great about your question and my subsequent research.
I was previously assuming that only the first was true and the second was false.
Then I found out that the first assumes geometry on a plane and the second
assumes some other non-plane basis.

This means that they are both equally true on the basis of their differing
foundations. Luckily I was already ready for that by restricting the notion
of true to a specific formal system they can both be equally true within
their respective formal system.

[wtf]

That was what was great about your question and my subsequent research.
I was previously assuming that only the first was true and the second was false.
Then I found out that the first assumes geometry on a plane and the second
assumes some other non-plane basis.

This means that they are both equally true on the basis of their differing
foundations. Luckily I was already ready for that by restricting the notion
of true to a specific formal system they can both be equally true within
their respective formal system.

No, you are misunderstanding.

They are both consistent; but not with each other. They are inconsistent with each other. Only one can be true about the world. They can NOT be equally true. But the axioms themselves can't tell you which. Only empirical science can do that.

But thanks, yes it is a great question. It shows that an axiomatic system can be consistent yet not true.

... the first assumes geometry on a plane and the second assumes some other non-plane basis.

And which is true of the universe in which we find ourselves? Isn't that a question for physicists to determine using empirical science? Logic can't help us.

[PeteOlcott]

That was what was great about your question and my subsequent research.
I was previously assuming that only the first was true and the second was false.
Then I found out that the first assumes geometry on a plane and the second
assumes some other non-plane basis.

This means that they are both equally true on the basis of their differing
foundations. Luckily I was already ready for that by restricting the notion
of true to a specific formal system they can both be equally true within
their respective formal system.

No, you are misunderstanding.

They are both consistent; but not with each other. They are inconsistent with each other. Only one can be true about the world. They can NOT be equally true. But the axioms themselves can't tell you which. Only empirical science can do that.

But thanks, yes it is a great question. It shows that an axiomatic system can be consistent yet not true.

One is true on a plane and the other is true on a sphere.
On a sphere is triangle actually has more than 180 degrees.
http://mathworld.wolfram.com/SphericalTriangle.html

This is fudging with truth a little bit because an actual triangle is only on a plane.
A spherical triangle would actually be an object that is comparable to a plane triangle
yet on a spherical surface rather than a plane.

So I guess for this same sort of reasoning parallel lines do not really meet.

[wtf]

One is true on a plane and the other is true on a sphere.
On a sphere is triangle actually has more than 180 degrees.
http://mathworld.wolfram.com/SphericalTriangle.html

This is fudging with truth a little bit because an actual triangle is only on a plane.
A spherical triangle would actually be an object that is comparable to a plane triangle
yet on a spherical surface rather than a plane.

So I guess for this same sort of reasoning parallel lines do not really meet.

First, let's note that when you say plane, you mean Euclidean space in general. It could be a cube or a 4-cube or 5-cube or n-cube. But it's "flat."

And spheres are only a special case of non-Euclidean geometry. There are saddle shapes, torii (n-dimensional donuts), etc. And some of those can be flat too!! This subject is differential geometry. There are many wild shapes out there.

And it can be tricky. A 2-torus (a standard donut shape) is actually flat, for technical reasons. So "flat" isn't really the word. But Euclidean versus non-Euclidean is a simple dichotomy sufficient for our needs.

But the point is that we happen to find ourselves living in a universe. And that universe has SOME geometry. Which is it? Is it Euclidean or non-Euclidean? It must be one or the other and it can't be both.

You seem to be redefining what everyone else calls validity, with the word "truth." Perhaps that's your idea of a stipulative definition. If you define '5' to mean 4, then 2 + 2 = '5'. But that doesn't make 2 + 2 = 5. It only means that you changed the meaning of a standard symbol. That's perfectly legal. But it doesn't change the meaning of the underlying objects. It only means you express the same concepts using different symbols. Syntax and semantics again. Marks on paper on the one hand; and meaning on the other.

True is what's true in the world. Is the world Euclidean or non-Euclidean? That's a matter for empirical science. Logic and math can't answer the question.

Can I ask you a personal question? We're all smart about some things and ignorant of others so no judgment. But is it really true that you never heard about non-Euclidean geometry before? Or that Einstein's great mind-blowing revelation was that we DON'T necessarily live in a Euclidean world? Gravity distorts straight lines.

Historically it was non-Euclidean geometry that got mathematicians interested in foundations, leading to set theory and ‎Gödel. It was this very example that led to the study of axiomatic systems.

[PeteOlcott]

One is true on a plane and the other is true on a sphere.
On a sphere is triangle actually has more than 180 degrees.
http://mathworld.wolfram.com/SphericalTriangle.html

This is fudging with truth a little bit because an actual triangle is only on a plane.
A spherical triangle would actually be an object that is comparable to a plane triangle
yet on a spherical surface rather than a plane.

So I guess for this same sort of reasoning parallel lines do not really meet.

First, let's note that when you say plane, you mean Euclidean space in general. It could be a cube or a 4-cube or 5-cube or n-cube. But it's "flat."

And spheres are only a special case of non-Euclidean geometry. There are saddle shapes, torii (n-dimensional donuts), etc. And some of those can be flat too!! This subject is differential geometry. There are many wild shapes out there.

And it can be tricky. A 2-torus (a standard donut shape) is actually flat, for technical reasons. So "flat" isn't really the word. But Euclidean versus non-Euclidean is a simple dichotomy sufficient for our needs.

But the point is that we happen to find ourselves living in a universe. And that universe has SOME geometry. Which is it? Is it Euclidean or non-Euclidean? It must be one or the other and it can't be both.

You seem to be redefining what everyone else calls validity, with the word "truth." Perhaps that's your idea of a stipulative definition. If you define '5' to mean 4, then 2 + 2 = '5'. But that doesn't make 2 + 2 = 5. It only means that you changed the meaning of a standard symbol. That's perfectly legal. But it doesn't change the meaning of the underlying objects. It only means you express the same concepts using different symbols. Syntax and semantics again. Marks on paper on the one hand; and meaning on the other.

True is what's true in the world. Is the world Euclidean or non-Euclidean? That's a matter for empirical science. Logic and math can't answer the question.

Can I ask you a personal question? We're all smart about some things and ignorant of others so no judgment. But is it really true that you never heard about non-Euclidean geometry before? Or that Einstein's great mind-blowing revelation was that we DON'T necessarily live in a Euclidean world? Gravity distorts straight lines.

Historically it was non-Euclidean geometry that got mathematicians interested in foundations, leading to set theory and ‎Gödel. It was this very example that led to the study of axiomatic systems.

I mean Cartesian plane.
I know that there are different non-Euclidean geometries. I solve problems by making them concrete and I had concrete examples.

I am converting formal proofs to use the sound deductive inference model. https://www.iep.utm.edu/val-snd/
having connected sequence of valid deductions from true premises to a true conclusion.

I am staying away from from formalizing synthetic truth.
According to Hindu Maya reality is fictional thus truth does not actually exist in physical reality.
https://www.britannica.com/topic/maya-Indian-philosophy

In purely analytic geometry both Euclidean geometry and non-Euclidean geometry are true.
I have been pondering the nature of truth for 50 years and providing the only correct one.

(1) There are certain expression of language that are defined to be true
and connected together by deductive inference:
"dogs are mammals"
"mammals breath"
∴ "dogs breath"

Because of the problem of induction, brain-in-a-vat and five-minute ago hypothesis
https://en.wikipedia.org/wiki/Brain_in_a_vat
https://en.wikipedia.org/wiki/Omphalos_ ... hypothesis
https://stanford.library.sydney.edu.au/ ... n-problem/
nothing not fitting this model counts as true.

Another way to say this is that true statements are statements that are completely
verified as true entirely on the basis of the meaning of their words.

I did not know that non-Euclidean geometry had non-plane geometric objects as its
foundation. I thought that it simply assumed a known falsehood.

Triangles on a sphere are not really triangles because triangles are defined on a plane.
Parallel lines never meet because parallel lines are defined on a plane.

Geometric objects that are comparable to triangles and parallel lines can
be defined on a basis other than a plane.

It does not matter if physical reality disagrees with the above stipulated definitions.
Because they are stipulated they form a mutually self-defining tautology.
All of conceptual truth functions this way as long as it remains consistent.
Inconsistency forces a choice.

My stipulated axioms eliminate undecidability and incompleteness from all formal
systems that were previously insufficiently expressive to detect and reject
semantically unsound logic sentences.

Axiom(0) Stipulates this definition of Axiom:
Expressions of language defined to have the semantic value of Boolean True.
Provides the symbolic logic equivalent of true premises.

Stipulating this specification of True and False: (TRUE ↔ ⊤ ∧ FALSE ↔ ⊥)
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))
Thus stipulating that consequences are provable from axioms.

Stipulating that formal systems are Boolean:
Axiom(3) ∀F ∈ Formal_System ∀x ∈ Closed_WFF(F) (True(F,x) ∨ False(F,x))
Screens out semantically unsound sentences as not belonging to the formal system.

The following logic sentence is refuted on the basis of Axiom(3)
∃F ∈ Formal_System ∃x ∈ Closed_WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
There is no sentence G of Formal System F that is neither True nor False in F.
Thus refuting the 1931 Incompleteness Theorem.

[Logik]

All of conceptual truth functions this way as long as it remains consistent.
Inconsistency forces a choice.

You do know that "consistency" itself is an arbitrary choice/property for any data system, right?

https://en.wikipedia.org/wiki/CAP_theorem

[PeteOlcott]

All of conceptual truth functions this way as long as it remains consistent.
Inconsistency forces a choice.

You do know that "consistency" itself is an arbitrary choice/property for any data system, right?

https://en.wikipedia.org/wiki/CAP_theorem

I am referring to the law of non contradiction.

[Logik]

I am referring to the law of non contradiction.

It's not a law. It's an axiom/proposition. It fails to take the arrow of time into account.

https://repl.it/repls/BoringRedundantFormulas

Code: Select all

p and ( not p )
=> true

[PeteOlcott]

I am referring to the law of non contradiction.

It's not a law. It's an axiom/proposition. It fails to take the arrow of time into account.

https://repl.it/repls/BoringRedundantFormulas

Code: Select all

p and ( not p )
=> true

IT DOES NOT FAIL TO TAKE THE ARROW OF TIME INTO ACCOUNT.
Default reasoning concludes the time is now unless otherwise specified.
None-the-less: ¬(True ↔ False) in all possible worlds at every point in time.

[Logik]

IT DOES NOT FAIL TO TAKE THE ARROW OF TIME INTO ACCOUNT.
Default reasoning concludes the time is now unless otherwise specified.

I have no idea what you mean by "now".

Formalize it for me. Is it an instant or an interval? Is time an Integer or a Real ?

None-the-less: ¬(True ↔ False) in all possible worlds at every point in time.

Unless, superposition

[PeteOlcott]

IT DOES NOT FAIL TO TAKE THE ARROW OF TIME INTO ACCOUNT.
Default reasoning concludes the time is now unless otherwise specified.

I have no idea what you mean by "now".

Formalize it for me. Is it an instant or an interval? Is time an Integer or a Real ?

None-the-less: ¬(True ↔ False) in all possible worlds at every point in time.

Unless, superposition

Synthetic "truth" is off topic:
The ONLY thing that we can know for certain about reality is that it seems to be a
continuous stream of physical sensations. Everything else about reality is pure speculation.

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. When the
term already exists, this definition may, but does not necessarily, contradict the dictionary
(lexical) definition of the term.

The only way that we can know anything regarding analytic truth is that some finite string
expressions of language are defined to have the semantic property of Boolean True.

It is stipulated that:
¬(True ↔ False) is one of those finite strings.

It is stipulated that:
Not(Living-Thing->Animal->Dog [Is-a-Type_of] Office_Building)

It is stipulated that:
"5 > 3"

Every element of all of analytic truth is essentially stipulated as true and it
it only the stipulation that makes it true it is not true for any other reason.

[Impenitent]

Every element of all of analytic truth is essentially stipulated as true and it
it only the stipulation that makes it true it is not true for any other reason.

nice circles.

as I said, defining the future (which is beyond perception - not currently existing) does not ensure its conformity to your definition...

-Imp