Post by: talanum1 on 16/04/2022 12:44:56
Here is a more up to date version:
Proof of Axioms of Propositional Logic wo name.pdf (403.18 kB - downloaded 59 times)
Post by: Bored chemist on 16/04/2022 12:48:53
No.
At best, all you have done is choose a different set of axioms.
Post by: talanum1 on 16/04/2022 13:03:18
Post by: Bored chemist on 16/04/2022 13:07:29
My axioms are more basic. It took creativity to devise.More basic than "and"?
Post by: talanum1 on 16/04/2022 13:13:10
More basic than "and"?
Yes. The idea of chopped up relations doesn't even arise in ordinary language.
Post by: Bored chemist on 16/04/2022 13:28:00
Which proves that it is not, actually, more basic.More basic than "and"?
Yes. The idea of chopped up relations doesn't even arise in ordinary language.
By the way, would it help if I actually read your PDF, or would I just spot lots of errors and inconsistencies?
Post by: talanum1 on 16/04/2022 13:31:23
Post by: Bored chemist on 16/04/2022 13:56:02
Don't waste people's time on stuff you know is faulty- that's just rude.
Post by: talanum1 on 16/04/2022 14:03:12
Post by: Bored chemist on 16/04/2022 14:06:45
Let me know when you think it's fixed.
Don't waste people's time on stuff you know is faulty- that's just rude.
Post by: talanum1 on 16/04/2022 14:25:46
[ Invalid Attachment ]
Post by: talanum1 on 16/04/2022 16:53:49
Post by: talanum1 on 17/04/2022 12:09:44
Post by: alancalverd on 17/04/2022 19:02:41
I have devised some axioms to prove "axioms" of Propositional Logic.An axiom by definition cannot be proved.
Post by: talanum1 on 18/04/2022 09:04:45
An axiom by definition cannot be proved.
That's why I put it in quotation marks.
Post by: Bored chemist on 19/04/2022 17:45:37
So... you knew it was wrong, but you said it anyway, and pretended that putting it in quotes made it OK.An axiom by definition cannot be proved.
That's why I put it in quotation marks.
How did you come to the conclusion that saying something, which you knew was wrong, was helpful?
Are you hoping to be prime minister?
Post by: talanum1 on 21/04/2022 10:22:19
So... you knew it was wrong, but you said it anyway, and pretended that putting it in quotes made it OK.
How did you come to the conclusion that saying something, which you knew was wrong, was helpful?
Are you hoping to be prime minister?
What are you talking about. They call it "axioms" but I proved it isn't axioms.
Post by: talanum1 on 30/04/2022 13:22:09
Something interesting: when there is more than one Attractor (in the thick of things i.e. when you need to apply an axiom more than once to a line) the axioms only allow approximate formulas to be derived, it's to be expected since the Attractors might interfere with each other.
Post by: talanum1 on 31/05/2022 14:11:42
Post by: Bored chemist on 31/05/2022 14:16:29
Comment on Bored chemist's: "you just chose some other axioms": I created a whole new sub-language!You missed the important bit when you quoted me.
At best, all you have done is choose a different set of axioms.
Post by: Origin on 31/05/2022 14:52:41
It is useful. I derive formulas with it.That would be interesting. Please derive a formula (any one is ok) using your new logic.
Thank you.
Post by: talanum1 on 01/06/2022 12:06:51
Post by: Origin on 01/06/2022 12:23:07
It is axioms no-one else would have found, because it requires pictures.Was that response to me?
All I asked is for you to give a quick example of this claim you made:
It is useful. I derive formulas with it.So, please derive a formula (any one is ok) using your new logic.
Post by: talanum1 on 01/06/2022 13:07:39
(~P, (P, ~Q) -> _ ) -> ~Q
where "~" means negation and "_" means the empty structure.
Proof:
line nr. Statement Reason
1 ~P, (P, ~Q) -> _ Premise
2 ~P, (P, ~Q) -> _ ->-( 1, A:AtI
3 (~P)->-( )->-(P ~Q) |->-(_)->-( 2, A:AD
4 (~P)->-( )->-(P ~Q) 3, A:EED
5 (~P)->-( )->-(P) )->-(~Q) 4, A:AM
6 ~(_) )->-(~Q) 5, A:AN
7 ~Q 6, A:NATL
where the reasons are axioms from the attached file and ~(_) is the empty structure negated (it always has truth value: "True"). "->" reads: "Therefore", ")-" reads: "Attractor" and "|-" reads "Stopper".
Post by: Bored chemist on 01/06/2022 13:09:21
Post by: talanum1 on 01/06/2022 13:21:09
Post by: Bored chemist on 01/06/2022 13:43:22
operation
introduced into
the empty structure
Always True Structure
Post by: Origin on 01/06/2022 14:06:13
What is Q?
What do the commas mean?
Big picture, what does the equation you are deriving mean?
Post by: talanum1 on 02/06/2022 09:07:47
Operation means: when you have a proposition P and apply the operation to it you get a different proposition Q. Of course just any such operation isn't useful in-that it can be used to prove formulas validly.
I can only express "introduction" by an axiom: if you introduce negation into a proposition P you get ~P.
"Empty structure" is the same as an empty page (no writing on it).
"Always True Structure" is any structure that is true no matter the truth values of its constituents.
Post by: talanum1 on 15/06/2022 17:15:45
Proof of "Axioms" of Propositional Logic:
Synopsis.
Willem F. Esterhuyse.
Abstract.
We introduce more basic axioms with which we are able to prove some
"axioms" of Propositional Logic. We use the symbols from my other article:
"Introduction to Logical Structures". Logical Structures (SrL) are graphs with
doubly labelled vertices with edges carrying symbols. The proofs are very
mechanical and does not require ingenuity to construct. It is easy to see that in
order to transform information, it has to be chopped up. Just look at a kid playing
with blocks with letters on them: he has to break up the word into letters to
assemble another word. Within SrL we take as our "atoms" propositions with
chopped up relations attached to them. We call the results: (incomplete)
"structures". We play it safe by allowing only relations among propositions to be
choppable. We will see whether this is the correct way of chopping up sentences
(it seems to be). This is where our Attractors (Repulsors) and Stoppers come in.
Attractors that face away from each other repels and so break a relation between
the two propositions. Then a Stopper attaches to the chopped up relation to
indicate it can't reconnect. So it is possible to infer sentences from sentences. The
rules I stumbled upon, to implement this, seems to be consistent. Sources differ
asto the axioms they choose but some of the most famous "axioms" are proved.
Modus Ponens occurs in all systems.
1. Introduction.
We use new operators called "Attractors" and "Stoppers". An Attractor ( symbol:
"-(" OR ")-") is an edge with a half circle symbol, that can carry any relation
symbol. Axioms for Attractors include A:AA (Axiom: Attractor Annihilation)
where we have as premise two structures named B with Attractors carrying the
"therefore" symbol facing each other and attached to two neighbouring structures:
B. Because the structures are the same and the Attractors face each other, and the
therefore symbols point in the same direction, they annihilate the structures B and
we are left with a conclusion of the empty structure. Like in:
((B)->-( )->-(B)) <-> (Empty Structure).
where "<->" means: "is equivalent to" or "follows from and vice vesa".
We also have the axiom: A:AtI (Attractor Introduction) in which we have a row
of structures as premise and conclusion of the same row of structures each with an
Attractor attached to them and pointing to the right or left. Like in:
A B C D <-> (A)-( (B)-( (C)-( (D)-(
OR:
A B C D <-> )-(A) )-(B) )-(C) )-(D)
A:AD distributes the Attractors and cut relations and places a Stopper on the
cut relation (see line 3 below). Stopper = "|-" or "-|".
Further axioms are: A:SD says that we may drop a Stopper at either end of a line.
And A:ASS says we can exchange Stoppers for Attractors (and vice versa) in a
line of structures as long as we replace every instance of the operators. A:AL says
we can link two attractors pointing trowards each other and attached to two
different structures.
We can prove: P OR P -> P. We prove Modus Ponens as follows:
Line nr. Statement Reason
1 B B -> C Premise
2 (B)->-( (B -> C)->-( 1, A:AtI
3 (B)->-( )->-(B) |->-(C)->-( 2, A:AD
4 |->-(C)->-( 3, A:AA
5 (C)->-( 4, A:SD
6 (C)->-| 5, A:ASS
7 C 6, A:SD
We see that the Attractors cuts two structures into three (line 2 to line 3).
We can prove AND-elimination, AND-introduction and transposition. We prove
Theorem: AND introduction (T:ANDI):
1 A B Premise
2 A -(x)-( B -(x)-( 1, A:AtI
3 (A)-(x)-| (B)-(x)-| 2, A:ASS
4 (A)-(x)-| B 3, A:SD
5 (A)-(x)-( B 4, A:ASS
6 (A)-(x)-(B) 5, T:AL
where "-(x)-" = "AND", and T:AL is a theorem to be proved by reasoning
backwards through:
1 A -(x)- B Premise
2 A -(x)- B -(x)-( 1, A:AtI
3 )-(x)-(A) |-(x)-(B)-(x)-( 2, A:AD
4 |-(x)-(A) )-(x)-(B)-(x)-| 3, A:ASS
5 A )-(x)-(B) 4, A:SD.
where the mirror image of this is proved similarly (by choosing to place the
Stopper on the other side of "-(x)-").
Modus Tollens and Syllogism can also be proven with these axioms.
We prove: Theorem (T:O): (A OR A) -> A:
1 A -(+)- A Premise
2 A -(+)- A -(+)-( 1, A:AtI
3 )-(+)-(A) |-(+)-(A)-(+)-( 2, A:AD
4 |-(+)-(A) )-(+)-(A)-(+)-| 3, A:ASS
5 A )-(+)-(A) 4, A:SDx2
6 A |-(+)-(A) 5, A:ASS
and from this (on introduction of a model taking only structures with truth tables
as real) we can conclude that A holds as required.
We prove Syllogism:
1 A -> B B -> C Premise
2 (A -> B)->-( (B -> C)->-( 1, A:AtI
3 )->-(A)->-| (B)->-( )->-(B) |->-(C)->-( 2, A:ADx2
4 (A)->-| (B)->-( )->-(B) |->-(C) 3, A:ASS, A:SDx2, A:ASS
5 (A)->-| |->-(C) 4, A:AA
6 (A)->-( )->-(C) 5, A:ASS
7 A -> C 6, A:AL
Post by: alancalverd on 16/06/2022 18:19:14
It is axioms no-one else would have found, because it requires pictures.Therefore it isn't an axiom. Call it whatever you like, but you'll confuse yourself if you misuse a technical term.
Post by: Bored chemist on 16/06/2022 18:51:22
2 months on, nobody has suggested a use for it.
Post by: talanum1 on 17/06/2022 10:30:12
Therefore it isn't an axiom. Call it whatever you like, but you'll confuse yourself if you misuse a technical term.
Just because no one else can find it does not mean it is not an axiom. This is not a popularity contest nor does it depend on someone's authority. It hangs or falls on the soundness of the logic.
Is this New Logic Usefull?
2 months on, nobody has suggested a use for it.
I have shown that you can derive formulas with it. It is like mathematics: does not necessarily have a practical application.
Post by: talanum1 on 17/06/2022 10:39:27
Proof of "Axioms" of Propositional Logic:
Synopsis.
Willem F. Esterhuyse.
Abstract.
We introduce more basic axioms with which we are able to prove some
"axioms" of Propositional Logic. We use the symbols from my other article:
"Introduction to Logical Structures". Logical Structures (SrL) are graphs with
doubly labelled vertices with edges carrying symbols. The proofs are very
mechanical and does not require ingenuity to construct. It is easy to see that in
order to transform information, it has to be chopped up. Just look at a kid playing
with blocks with letters on them: he has to break up the word into letters to
assemble another word. Within SrL we take as our "atoms" propositions with
chopped up relations attached to them. We call the results: (incomplete)
"structures". We play it safe by allowing only relations among propositions to be
choppable. We will see whether this is the correct way of chopping up sentences
(it seems to be). This is where our Attractors (Repulsors) and Stoppers come in.
Attractors that face away from each other repels and so break a relation between
the two propositions. Then a Stopper attaches to the chopped up relation to
indicate it can't reconnect. So it is possible to infer sentences from sentences. The
rules I stumbled upon, to implement this, seems to be consistent. Sources differ
asto the axioms they choose but some of the most famous "axioms" are proved.
Modus Ponens occurs in all systems.
1. Introduction.
We use new operators called "Attractors" and "Stoppers". An Attractor ( symbol:
"-(" OR ")-") is an edge with a half circle symbol, that can carry any relation
symbol. Axioms for Attractors include A:AA (Axiom: Attractor Annihilation)
where we have as premise two structures named B with Attractors carrying the
"therefore" symbol facing each other and attached to two neighbouring structures:
B. Because the structures are the same and the Attractors face each other, and the
therefore symbols point in the same direction, they annihilate the structures B and
we are left with a conclusion of the empty structure. Like in:
((B)->-( )->-(B)) <-> (Empty Structure).
where "<->" means: "is equivalent to" or "follows from and vice vesa".
We also have the axiom: A:AtI (Attractor Introduction) in which we have a row
of structures as premise and conclusion of the same row of structures each with an
Attractor attached to them and pointing to the right or left. Like in:
A B C D <-> (A)-( (B)-( (C)-( (D)-(
OR:
A B C D <-> )-(A) )-(B) )-(C) )-(D)
A:AD distributes the Attractors and cut relations and places a Stopper on the
cut relation (see line 3 below). Stopper = "|-" or "-|".
Further axioms are: A:SD says that we may drop a Stopper at either end of a line.
And A:ASS says we can exchange Stoppers for Attractors (and vice versa) in a
line of structures as long as we replace every instance of the operators. A:AL says
we can link two attractors pointing trowards each other and attached to two
different structures.
We can prove: P OR P -> P. We prove Modus Ponens as follows:
Line nr. Statement Reason
1 B B -> C Premise
2 (B)->-( (B -> C)->-( 1, A:AtI
3 (B)->-( )->-(B) |->-(C)->-( 2, A:AD
4 |->-(C)->-( 3, A:AA
5 (C)->-( 4, A:SD
6 (C)->-| 5, A:ASS
7 C 6, A:SD
We see that the Attractors cuts two structures into three (line 2 to line 3).
We can prove AND-elimination, AND-introduction and transposition. We prove
Theorem: AND introduction (T:ANDI):
1 A B Premise
2 A -(x)-( B -(x)-( 1, A:AtI
3 (A)-(x)-| (B)-(x)-| 2, A:ASS
4 (A)-(x)-| B 3, A:SD
5 (A)-(x)-( B 4, A:ASS
6 (A)-(x)-(B) 5, T:AL
where "-(x)-" = "AND", and T:AL is a theorem to be proved by reasoning
backwards through:
1 A -(x)- B Premise
2 A -(x)- B -(x)-( 1, A:AtI
3 )-(x)-(A) |-(x)-(B)-(x)-( 2, A:AD
4 |-(x)-(A) )-(x)-(B)-(x)-| 3, A:ASS
5 A )-(x)-(B) 4, A:SD.
where the mirror image of this is proved similarly (by choosing to place the
Stopper on the other side of "-(x)-").
Modus Tollens and Syllogism can also be proven with these axioms.
We prove: Theorem (T:O): (A OR A) -> A:
1 A -(+)- A Premise
2 A -(+)- A -(+)-( 1, A:AtI
3 )-(+)-(A) |-(+)-(A)-(+)-( 2, A:AD
4 |-(+)-(A) )-(+)-(A)-(+)-| 3, A:ASS
5 A )-(+)-(A) 4, A:SDx2
6 A |-(+)-(A) 5, A:ASS
and from this (on introduction of a model taking only structures with truth tables
as real) we can conclude that A holds as required.
We prove Syllogism:
1 A -> B B -> C Premise
2 (A -> B)->-( (B -> C)->-( 1, A:AtI
3 )->-(A)->-| (B)->-( )->-(B) |->-(C)->-( 2, A:ADx2
4 (A)->-| (B)->-( )->-(B) |->-(C) 3, A:ASS, A:SDx2, A:ASS
5 (A)->-| |->-(C) 4, A:AA
6 (A)->-( )->-(C) 5, A:ASS
7 A -> C 6, A:AL
Post by: Bored chemist on 17/06/2022 12:12:30
It is like mathematics: does not necessarily have a practical application.You just answered your own question.
Post by: Origin on 17/06/2022 16:45:01
Post by: talanum1 on 19/06/2022 09:58:49
How many threads of this junk do you need?
I have to make it known to mainstream scientists. I won't post any more threads of it.
You call it junk because you can't follow it.
Post by: Bored chemist on 19/06/2022 10:14:50
I have to make it known to mainstream scientists.Why?
Do you not realise that they have better things to do?
You call it junk because you can't follow it.Nobody did.
Post by: talanum1 on 19/06/2022 14:40:03
I have to make it known to mainstream scientists.
Why?
Do you not realise that they have better things to do?
Maybe there is a Logician out there who finds it interesting. It is of interest because it extends Language in the more basic direction: something no one else has yet managed to do, and because its proofs are mechanical. A computer can be programmed to derive formulas with it.
You call it junk because you can't follow it.
Nobody did.
Why, it's easy to follow?
Post by: talanum1 on 10/08/2022 13:25:44
Now you would have to deal with it.
Post by: Bored chemist on 10/08/2022 16:07:23
Is this New Logic Usefull?And now 4 months...
2 months on, nobody has suggested a use for it.
Post by: Origin on 10/08/2022 16:53:33
Ha ha. The Logic has been printed by IOSR Journal of Mathematics at: https://www.iosrjournals.org/iosr-jm/papers/Vol18-issue4/Ser-3/D1804032041.pdf.
Now you would have to deal with it.
Paying somebody to publish your silliness on the internet for $75 bucks is meaningless.
IOSR Journals are listed as 'Predatory Journals'. That means they don't actually peer review anything and just take your money to put your paper on the internet.
Post by: Bored chemist on 10/08/2022 18:02:28
Post by: talanum1 on 26/09/2022 11:17:24
Post by: Bored chemist on 26/09/2022 22:24:08
It is usefull. I ran the Logic in mind and this led me to discover the places in mind that language starts. Now I think pages.Didi that make sense when it was in your head?
Post by: talanum1 on 27/09/2022 09:45:46
Post by: talanum1 on 02/10/2022 09:33:51