G asserts its own unprovability in F

[Skepdick]

An axiom is proven true on the basis of it being an axiom.

It's not proven true.
It's defined true.

Do you even understand the semantic difference between a definition and a proof?

[PeteOlcott]

An axiom is proven true on the basis of it being an axiom.

It's not proven true.
It's defined true.

Do you even understand the semantic difference between a definition and a proof?

That is it defined true proves that it is true otherwise we have no way to
prove that cats are animals.

When proofs begin with true premises then provable(p,x) entails true(p,x).
When proofs begin with false premises then provable(p,x) DOES NOT entail true(p,x).

[Skepdick]

That is it defined true proves that it is true otherwise we have no way to
prove that cats are animals.

When proofs begin with true premises then provable(p,x) entails true(p,x).
When proofs begin with false premises then provable(p,x) DOES NOT entail true(p,x).

You continue to conflate definitions and proofs.

It's like you refuse to acknowledge the semantic difference between an entailment of an empty set and an entailment of a non-empty set.

[PeteOlcott]

That is it defined true proves that it is true otherwise we have no way to
prove that cats are animals.

When proofs begin with true premises then provable(p,x) entails true(p,x).
When proofs begin with false premises then provable(p,x) DOES NOT entail true(p,x).

You continue to conflate definitions and proofs.

It's like you refuse to acknowledge the semantic difference between an entailment of an empty set and an entailment of a non-empty set.

Mendelson says that entailment from an empty set is encoded as ⊢𝒞
meaning provable on the basis of the empty set of premises.
According to Haskell Curry this also means true by definition.

The only big difference between provable and true is that some expressions
that are provable are untrue.

Gödel has convinced many people that an expression can be true and unprovable.
This is impossible. He is merely pulling this trick:

This sentence is not true: "This sentence is not true"
The outer sentence is true because the inner sentence is not a truth bearer.

[Skepdick]

Mendelson says that entailment from an empty set is encoded as ⊢𝒞
meaning provable on the basis of the empty set of premises.
According to Haskell Curry this also means true by definition.

The encoding or the representation are immaterial.

There is a semantic difference. Are you so blind that you don't see it? Or do you profusely refuse to see it?


A definition takes no inputs.

Code: Select all

char* definition(void);

A proof does.

Code: Select all

char* proof(char premises[]);

The only big difference between provable and true is that some expressions
that are provable are untrue.

That's not even a difference. You can define things as untrue also.

[PeteOlcott]

Mendelson says that entailment from an empty set is encoded as ⊢𝒞
meaning provable on the basis of the empty set of premises.
According to Haskell Curry this also means true by definition.

The only big difference between provable and true is that some expressions
that are provable are untrue.

The encoding or the representation are immaterial.

There is a semantic difference. Are you so blind that you don't see it? Or do you profusely refuse to see it?

In other words you don't understand that when some untrue expressions
are proven that this is semantically different than when some true
expressions are proven.

We can prove that the Moon is made from cheese on the basis of the premise that the Moon is made from green cheese.

[Skepdick]

In other words you don't understand that when some untrue expressions
are proven that this is semantically different than when some true
expressions are proven.

So you don't even understand formal semantics; or the type-signature of a proof.

Great.

[PeteOlcott]

In other words you don't understand that when some untrue expressions
are proven that this is semantically different than when some true
expressions are proven.

So you don't even understand formal semantics.

Great.

An empty claim with no basis

[Skepdick]

In other words you don't understand that when some untrue expressions
are proven that this is semantically different than when some true
expressions are proven.

So you don't even understand formal semantics.

Great.

An empty claim with no basis

⊢ You don't even understand formal semantics.

Q.E.D

[PeteOlcott]

In other words you don't understand that when some untrue expressions
are proven that this is semantically different than when some true
expressions are proven.

So you don't even understand formal semantics; or the type-signature of a proof.

Great.

The entire body of analytical truth is simply expressions of language
that are semantically derived from expressions of language that have
been stipulated to be true.

When the semantics is correctly specified syntactically then expressions
of language can be syntactically derived from expressions of language
that have been stipulated to be true.

The latter is the essence of formal semantics

[Skepdick]

When the semantics is correctly specified syntactically then expressions
of language can be syntactically derived from expressions of language
that have been stipulated to be true.

What does that even mean?

Syntax is not semantics.
Syntax is syntax
Semantics is semantics.

They are different.

https://www.tutorialspoint.com/differen ... -semantics

[PeteOlcott]

When the semantics is correctly specified syntactically then expressions
of language can be syntactically derived from expressions of language
that have been stipulated to be true.

What does that even mean?

Syntax is not semantics.
Syntax is syntax
Semantics is semantics.

So you had no idea that semantics can be expressed syntactically.
Your ignorance is not my error. The term "formal" of formal semantics means that semantics has been formalized syntactically.

[Skepdick]

So you had no idea that semantics can be expressed syntactically.
Your ignorance is not my error.

You are 100% correct. Your ignorance is your error.

Semantics is not fully expressable in syntax.

[PeteOlcott]

So you had no idea that semantics can be expressed syntactically.
Your ignorance is not my error.

You are 100% correct. Your ignorance is your error.

Semantics is not fully expressable in syntax.

The formal language semantics of formal languages merely requires an
algorithm to encode this.

To express natural language semantics syntactically required ChatGPT
to automate the process. The CYC project spent 700 labor years trying
to manually formalize the subset of knowledge known as common sense.

Formalizing all of natural language semantics merely involves specifying
relations between finite strings. It is more effective and efficient to use
GUIDs as placeholders for the unique sense meanings of words.

[Skepdick]

The formal language semantics of formal languages merely requires an
algorithm to encode this.

And the input of this algorithm is ...?