1

New Theories / Is the Consistency Condition for Logics Provable?

« on: Yesterday at 10:19:04 »

Is it provable that a system of Logic is consistent iff there is a true formula of the system that cannot be proved in the system? If yes please provide a reference.

2

New Theories / Does Recursive Functions lead to Contradiction?

« on: 01/08/2022 09:40:19 »

Principia Mathematica says a function f cannot be defined in terms of itself - it leads to contradiction. This contradicts mathematics, which allow recursive functions.

3

New Theories / Re: Is Mathematics Unintuitive?

« on: 28/07/2022 08:13:09 »

But infinitely thin means invisible or worse: non-existing - unintuitive.

4

New Theories / Re: Is Mathematics Unintuitive?

« on: 21/07/2022 09:09:17 »

Take a line on a piece of paper. Mathematics says a line is infinitessimaly thin. However our experience says a line has finite thickness otherwise it would be invisible. Also our physics says there is a molecule next to the line.

5

New Theories / Re: Is Classical Logic Unsound?

« on: 08/07/2022 12:12:54 »

I didn't know A -> B needs to be the whole truth.

6

New Theories / Is Classical Logic Unsound?

« on: 07/07/2022 09:23:23 »

Yes. In Classical Logic Syllogism: (A -> B, B -> ~C) -> (A -> ~C) is provable. However I found an interpretation that makes the premises true and the conclusion false: take A = "Are mortal", B = "All men" and C = "are dogs", "~" = not. Thus it does not semantically follow.

7

New Theories / Is Mathematics Unintuitive?

« on: 04/07/2022 10:14:54 »

It is because there is no real number next to zero. However in real life: take a line on a tiled floor. We know there is molecules right next to the line and they do not tend to zero. If this wasn't so there would be no tiling of floors

8

New Theories / Re: Is this New Logic Usefull?

« 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?

9

New Theories / Re: Is this New Logic Usefull?

« 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.

10

New Theories / Is this New Logic Useful?

« on: 17/06/2022 10:39:27 »

Here is the logic:

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

11

New Theories / Re: Is this New Logic Usefull?

« 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.

12

New Theories / Re: Is this New Logic Usefull?

« on: 15/06/2022 17:15:45 »

Here is a Synopsis:

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

13

General Science / Is Mathematics Inconsistent?

« on: 06/06/2022 12:12:26 »

It is: we have the Banach-Tarski Paradox which says that one sphere is two spheres or 1 = 2 or 1 ~= 1. One can also prove 1 = 1. So here is a inconsistency!

14

New Theories / Re: Is this New Logic Usefull?

« on: 02/06/2022 09:07:47 »

P and Q are structures (but you can also take them as propositions), "," is exist together. The formula hasn't got an easy meaning: you would be hard pressed to encounter the formula in real life.

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.

15

New Theories / Re: Is this New Logic Usefull?

« on: 01/06/2022 13:21:09 »

Negation is the operation that: when introduced into the empty structure produces an "Always True Structure".

16

New Theories / Re: Is this New Logic Usefull?

« on: 01/06/2022 13:07:39 »

I can derive:

(~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".

17

New Theories / Re: Is this New Logic Usefull?

« on: 01/06/2022 12:06:51 »

It is axioms no-one else would have found, because it requires pictures.

18

New Theories / Re: Is this New Logic Usefull?

« on: 31/05/2022 14:11:42 »

Comment on Bored chemist's: "you just chose some other axioms": I created a whole new sub-language!

19

New Theories / Do You Find this New Logic Interesting?

« on: 17/05/2022 09:31:13 »

Here is the Logic:

I discovered a new non-trivial axiom. See Structure 2.0.10.9. I added a few theorems:


Knowledge_Organization 4 wo name.pdf (1125.74 kB - downloaded 14 times)

20

New Theories / Re: Is ζ(-1) = -1/12 Inconsistent?

« on: 13/05/2022 13:24:50 »

We have ς (s) only defined for Re(s) > 1, and analytic continuation says: assign some values for ς(s) for Re (s) <= 1 that analytically follows from it's value for Re (s) > 1. These values are wholly determined from it's values for Re(s) > 1, so they need not be the same as expected from the definition of ς (s). We must take these values as an (inconvenient) fiction.