Philosophy Now Forum

PeteOlcott wrote: ↑Sun Apr 02, 2023 10:26 pm When we understand the underlying semantics of provability is a sequence
of inference steps deriving x then we know Provable(x) is not merely a
meaningless propositional variable.

When x is proved from a sequence of inference steps and this same set of
inference steps proves that x is true: Provable(x) → True(x) then True(x) and
~Provable(x) is impossible.

Literally every axiom satisfies True and ~Provable

If an axiom is Provable then it's inferable from some other axiom.
Rinse.
Repeat.

Till you arrive at some axiom that you are willing to accept as true, but you can't prove it because it's not inferable from any prior axiom.

Skepdick wrote: ↑Sun Apr 02, 2023 10:47 pm

PeteOlcott wrote: ↑Sun Apr 02, 2023 10:26 pm When we understand the underlying semantics of provability is a sequence
of inference steps deriving x then we know Provable(x) is not merely a
meaningless propositional variable.

When x is proved from a sequence of inference steps and this same set of
inference steps proves that x is true: Provable(x) → True(x) then True(x) and
~Provable(x) is impossible.

Literally every axiom satisfies True and ~Provable

If an axiom is Provable then it's inferable from some other axiom.
Rinse.
Repeat.

Till you arrive at some axiom that you are willing to accept as true, but you can't prove it because it's not inferable from any prior axiom.

Axioms are not true and unprovable, they are provable on the basis that they are axioms.
See also Mendelson.

PeteOlcott wrote: ↑Sun Apr 02, 2023 10:59 pm Axioms are not true and unprovable, they are provable on the basis that they are axioms.

Does there ever come a point in your reasoning where you go "Oh, fuck! Yeah. I am an idiot!"?

Axioms are not provable. They are assumed/declared/defined as true. That's not a proof because it contains no inference steps of any sort.

Or if you want to be a wise ass then here is an axiom (which is provable on the basis that it's an axiom): This sentence is false.

Skepdick wrote: ↑Sun Apr 02, 2023 11:05 pm

PeteOlcott wrote: ↑Sun Apr 02, 2023 10:59 pm Axioms are not true and unprovable, they are provable on the basis that they are axioms.

Does there ever come a point in your reasoning where you go "Oh, fuck! Yeah. I am an idiot!"?

Axioms are not provable. They are assumed/declared/defined as true. That's not a proof because it contains no inference steps of any sort.

Or if you want to be a wise ass then here is an axiom (which is provable on the basis that it's an axiom): This sentence is false.

Mendelson disagrees. He stipulated that theorems are distinguished as provable from an empty set of premises ⊢C

PeteOlcott wrote: ↑Sun Apr 02, 2023 11:12 pm

Skepdick wrote: ↑Sun Apr 02, 2023 11:05 pm

PeteOlcott wrote: ↑Sun Apr 02, 2023 10:59 pm Axioms are not true and unprovable, they are provable on the basis that they are axioms.

Does there ever come a point in your reasoning where you go "Oh, fuck! Yeah. I am an idiot!"?

Axioms are not provable. They are assumed/declared/defined as true. That's not a proof because it contains no inference steps of any sort.

Or if you want to be a wise ass then here is an axiom (which is provable on the basis that it's an axiom): This sentence is false.

Mendelson disagrees. He stipulated that theorems are distinguished as provable from an empty set of premises ⊢C

That’s literally the principle of explosion.

ex falso sequitur quodlibet
From falsify (the empty set) anything follows!

Skepdick wrote: ↑Sun Apr 02, 2023 11:16 pm

PeteOlcott wrote: ↑Sun Apr 02, 2023 11:12 pm

Skepdick wrote: ↑Sun Apr 02, 2023 11:05 pm
Does there ever come a point in your reasoning where you go "Oh, fuck! Yeah. I am an idiot!"?

Axioms are not provable. They are assumed/declared/defined as true. That's not a proof because it contains no inference steps of any sort.

Or if you want to be a wise ass then here is an axiom (which is provable on the basis that it's an axiom): This sentence is false.

Mendelson disagrees. He stipulated that theorems are distinguished as provable from an empty set of premises ⊢C

That’s literally the principle of explosion.

ex falso sequitur quodlibet
From falsify (the empty set) anything follows!

That cats are animals is proven by the fact that cats <are> animals.
"cats are animals" ∈ (a)

Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
T is a subset of (a)

True(T,X) means X ∈ (a) or T ⊨□ X

PeteOlcott wrote: ↑Sun Apr 02, 2023 11:27 pm That cats are animals is proven by the fact that cats <are> animals.
"cats are animals" ∈ (a)

That’s not a proof. That’s an axiom.

That cats are animals is true but not provable.

Q.E.D

Skepdick wrote: ↑Sun Apr 02, 2023 11:34 pm

PeteOlcott wrote: ↑Sun Apr 02, 2023 11:27 pm That cats are animals is proven by the fact that cats <are> animals.
"cats are animals" ∈ (a)

That’s not a proof. That’s an axiom.

That cats are animals is true but not provable.

Q.E.D

It is provable on the basis that it is stipulated to be true.

PeteOlcott wrote: ↑Sun Apr 02, 2023 11:41 pm

Skepdick wrote: ↑Sun Apr 02, 2023 11:34 pm

PeteOlcott wrote: ↑Sun Apr 02, 2023 11:27 pm That cats are animals is proven by the fact that cats <are> animals.
"cats are animals" ∈ (a)

That’s not a proof. That’s an axiom.

That cats are animals is true but not provable.

Q.E.D

It is provable on the basis that it is stipulated to be true.

Provable means “follows from the axioms”.

Which axioms does “cats are animals” follow from?

I believe Tarski and I are on the same page in certain respects. His views, are as usual, natural, as natural as trees are I suppose. There's a very good chance that we can say something nasty about Tarski, philosophically/logically, and his proof. This has always been so, is, always will be. I for one can't see why not Tarski and ... his ... proof? By the way, it isn't quite clear to me whether Tarski was simply commenting, remarking, or doing what some of you claim he's doing.

Skepdick wrote: ↑Mon Apr 03, 2023 6:08 am

PeteOlcott wrote: ↑Sun Apr 02, 2023 11:41 pm

Skepdick wrote: ↑Sun Apr 02, 2023 11:34 pm
That’s not a proof. That’s an axiom.

That cats are animals is true but not provable.

Q.E.D

It is provable on the basis that it is stipulated to be true.

Provable means “follows from the axioms”.

Which axioms does “cats are animals” follow from?

Although Tarski seems to mean follows from axioms,
Provable usually means follows from premises.
“cats are animals” is an axiom.

PeteOlcott wrote: ↑Mon Apr 03, 2023 2:34 pm Although Tarski seems to mean follows from axioms,
Provable usually means follows from premises.

Axioms. Premises.

They are functionally equivalent.

PeteOlcott wrote: ↑Mon Apr 03, 2023 2:34 pm “cats are animals” is an axiom.

Great! So. let x = “cats are animals"

Axiom(x) → ~Provable(x)
Axiom(x) → True(x)
∴ Axiom(x) → True(x) ∧ ~Provable(x)

Q.E.D

Agent Smith wrote: ↑Mon Apr 03, 2023 7:03 am I believe Tarski and I are on the same page in certain respects. His views, are as usual, natural, as natural as trees are I suppose. There's a very good chance that we can say something nasty about Tarski, philosophically/logically, and his proof. This has always been so, is, always will be. I for one can't see why not Tarski and ... his ... proof? By the way, it isn't quite clear to me whether Tarski was simply commenting, remarking, or doing what some of you claim he's doing.

You can read his actual two-page proof right here:
https://liarparadox.org/Tarski_275_276.pdf

I used to think that Tarski's Object language / Metalanguage was great when
the Object language is natural language such as English and the Meta language
is natural language formalized in something like higher order logic.
https://plato.stanford.edu/entries/tars ... #ObjLanMet

The following formal system does not need the Object Language / Meta language
dichotomy. Self contradictory expressions can simply be rejected as non-truth bearers
in the following system.

Introducing the foundation of correct reasoning

Just like with syllogisms conclusions a semantically necessary consequence of their premises

Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the semantic property of Boolean true.
Like Prolog Facts except every natural language expression can be encoded.

(b) Some expressions of language L are a semantically necessary consequence of others.
Like Prolog Rules except every natural language expression can be encoded.

P is a subset of expressions of language L // one or more elements of (a) and/or (b)
T is a subset of (a)

Provable(P,X) means P ⊨□ X
True(T,X) means X ∈ (a) or T ⊨□ X
False(T,X) means T ⊨□ ~X

Skepdick wrote: ↑Mon Apr 03, 2023 2:44 pm

PeteOlcott wrote: ↑Mon Apr 03, 2023 2:34 pm Although Tarski seems to mean follows from axioms,
Provable usually means follows from premises.

Axioms. Premises.

They are functionally equivalent.

PeteOlcott wrote: ↑Mon Apr 03, 2023 2:34 pm “cats are animals” is an axiom.

Great! So. let x = “cats are animals"

Axiom(x) → ~Provable(x)
Axiom(x) → True(x)
∴ Axiom(x) → True(x) ∧ ~Provable(x)

Q.E.D

Axiom(x) → ~Provable(x) It like saying I have a cup of water therefore my cup is empty.
Tarski simply contradicts himself. Line (5) proves that line (3) is wrong.

PeteOlcott wrote: ↑Mon Apr 03, 2023 2:55 pm Axiom(x) → ~Provable(x) It like saying I have a cup of water therefore my cup is empty.

Even God can't help this level of idiocy.

If "I have a cup of water" is an axiom it follows that "I have a cup of water" is not provable.
It's called non-provable because it doesn't appears on the right-hand side of the therefore.

??? therefore I have a cup of water.