Proof of "Axioms" of Propositional Logic.

My proof of Modus Ponens is no good: A:AA allows us to write: (B)->-( )->-(B) (C) -<>- (C). But then we can prove "false" -<>- "true" by and elimination. So I'm back to square 1.

We can however still prove "and elimination" and "and introduction" since they don't use this axiom.

I challenge anyone to come up with a proof of Modus Ponens - one that isn't circular like the usual "proof".
 
talanum1 said:
My proof of Modus Ponens is no good: ....
So I'm back to square 1.
Click to expand...

Aah, now you're sounding more like a careful scientist.

talanum1 said:
We can however still prove "and elimination" and "and introduction" since they don't use this axiom.

I challenge anyone to come up with a proof of Modus Ponens - one that isn't circular like the usual "proof".
Click to expand...

Hmm...
 
Of course all proofs in the paper using A:AA is no good.
 
We could just change A:AA to have the empty structure on RS, but then indicate that we reasoned through a logical singularity by appending: "->O". A:AA: ((B)->-( )->-(B))-<>-((_) ->O). Then ""false" ->O" evaluates to "true". We add the axiom A:TAT: true )->-(C) -<>-(C).

Then we change the MP proof to:

Line number Statement Reason
1 B B -> C Premise
2 (B)->-( (B -> C)->-( 1, A:AtI
3 (B)->-( )->-(B) []->-(C)->-( 2, A:AD
4 true []->-(C)->-( 3, A:AA
5 true )->-(C)->-[] 4, A:ASS
6 true )->-(C) 5, A:SD
7 C 6, A:TAT
 
We should have "true ->O" to evaluate to "false", and when checked against A-(+)-(X|)--(_) ->O) <> A-(+)-false <> A we see it must hold.
 
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I managed to prove another "axiom" of Propositional Logic:

We need two more axioms:

A:AAS: ( )-->--(B)-->--[] (A) (C)... []-->-- )-->--(B) (D) (E)...) <-> ((B)-->--[] (A) (C)...(D) (E)...)

where the Stopper is assumed attached to the second structure "B" and:

A:AO ((A)-->--( []-->-- []-->--(B)) <-> ((A)-->--( (B)).

Then we can prove the "axiom": (B-->--(C-->--D))--->--((B-->--C)-->--(B-->--D)) by reasoning backwards through:

Line # Statement Reason
1 (B-->--C)-->--(B-->--D) Premise
2 ((B-->--C)-->--(B-->--D))-->--( 1, A:AtI
3 )-->--(B-->--C) []-->--(B-->--D)-->--( 2, A:AD
4 )-->--(B)-->--[] (C)-->--( []-->--[ )-->--(B) []-->--(D)-->--( ] 3, A:AD
5 )-->--(B)-->--[] (C)-->--( []-->-- )-->--(B) []-->-- []-->--(D)-->--( 4, A:ASS, A:AM, A:ASS
6 (B)-->--[] (C)-->--( []-->-- []-->--(D)-->--( 5, A:AAS
7 (B)-->--[] (C)-->--( []-->-- []-->--(D) 6, A:ASS, A:SD, A:ASS
8 (B)-->--[] ((C)-->--( (D)) 7, A:AO
9 (B)-->--[] ((C)-->--(D)) 8, T:AL
10 (B)-->--( ((C)-->--(D)) 9. A:ASS
11 (B)-->--((C)-->--(D)) 10, T:AL
 
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talanum1 said:
I managed to prove another "axiom" of Propositional Logic:

We need two more axioms:

A:AAS: ( )-->--(B)-->--[] (A) (C)... []-->-- )-->--(B) (D) (E)...) <-> ((B)-->--[] (A) (C)...(D) (E)...)

where the Stopper is assumed attached to the second structure "B" and:

A:AO ((A)-->--( []-->-- []-->--(B)) <-> ((A)-->--( (B)).

Then we can prove the "axiom": (B-->--(C-->--D))--->--((B-->--C)-->--(B-->--D)) by reasoning backwards through:

Line # Statement Reason
1 (B-->--C)-->--(B-->--D) Premise
2 ((B-->--C)-->--(B-->--D))-->--( 1, A:AtI
3 )-->--((B-->--C) []-->--(B-->--D))-->--( 2, A:AD
4 )-->--(B)-->--[] (C)-->--( []-->--[ )-->--(B) []-->--(D)-->--( ] 3, A:AD
5 )-->--(B)-->--[] (C)-->--( []-->-- )-->--(B) []-->-- []-->--(D)-->--( 4, A:ASS, A:AM, A:ASS
6 (B)-->--[] (C)-->--( []-->-- []-->--(D)-->--( 5, A:AAS
7 (B)-->--[] (C)-->--( []-->-- []-->--(D) 6, A:ASS, A:SD, A:ASS
8 (B)-->--[] ((C)-->--( (D)) 7, A:AO
9 (B)-->--[] ((C)-->--(D)) 8, T:AL
10 (B)-->--( ((C)-->--( (D)) 9. A:ASS
11 (B)-->--((C)-->--( (D)) 10, T:AL
Click to expand...
Fk me this bs is still ongoing, has Mensa found out about this thread yet???
 
talanum1 said:
A:AAS: ( )-->--(B)-->--[] (A) (C)... []-->-- )-->--(B) (D) (E)...) <-> ((B)-->--[] (A) (C)...(D) (E)...)
...
5 )-->--(B)-->--[] (C)-->--( []-->-- )-->--(B) []-->-- []-->--(D)-->--( 4, A:ASS, A:AM, A:ASS
6 (B)-->--[] (C)-->--( []-->-- []-->--(D)-->--( 5, A:AAS
7 (B)-->--[] (C)-->--( []-->-- []-->--(D) 6, A:ASS, A:SD, A:ASS
....
10 (B)-->--( ((C)-->--( (D)) 9. A:ASS
Click to expand...

Quite a bit of ASS involved in this one :unsure:

images
 
@talanum1 's one thread was closed, it's nice that he has a belief system but it took it way beyond how formal science works.
This was the last post:

talanum1

Post in thread 'I can Prove Neutrons don't Decay by Quarks Transforming.'

God created the Universe by coupling a singularity to a computer and then running the program of the Universe on it. Thus the computer is now linked to all of space. This is how space can calculate i.e. "think".
  • Love

Probably if he posts more of it or in one of his existing dozen or so science forum threads, the axe will fall. Maybe the dude will go back to using the Off Topic forum with wider interests.
You are fun to talk to.
 
cat41 said:
@talanum1 's one thread was closed, it's nice that he has a belief system but it took it way beyond how formal science works.
This was the last post:

talanum1

Post in thread 'I can Prove Neutrons don't Decay by Quarks Transforming.'

God created the Universe by coupling a singularity to a computer and then running the program of the Universe on it. Thus the computer is now linked to all of space. This is how space can calculate i.e. "think".
  • Love

Probably if he posts more of it or in one of his existing dozen or so science forum threads, the axe will fall. Maybe the dude will go back to using the Off Topic forum with wider interests.
You are fun to talk to.
Click to expand...
Yep, nothing like logging on, on a fine Sunday to find you've been censored.
That thread was way overdue for being locked, but clearly the death of comedy is very real.
 
Brian_G said:
I'm curious, let's see what strange darkened alley he's going down now. Will have to use the dreaded AI though :whistling:








"Predictive Text":giggle: says;


Short version​


Your rewrite rules and truth tables don’t establish soundness or unsoundness because the symbols don’t yet have a defined meaning. Until semantics are fixed, the system can neither generate true nor false statements, so terms like “sound,” “unsound,” and “metalogical” don’t apply. Once meaning is added, the rewrite rules will either be valid inference rules or simply meta-level conveniences—but they won’t count as proofs of or against the axioms themselves.

Detailed reply (layman’s terms)​


What you’re calling “proof of axioms” mixes together two different levels of reasoning in logic:


  1. the object-level (the formal system itself), and
  2. the meta-level (the commentary about the formal system).

Once these two levels start getting blended, almost any symbol manipulation can be made to look “plausible,” but it no longer has a clear meaning.


1. “Unsound because it generates a false statement from a true one”​


In logic, an axiom isn’t “unsound” just because you can write down a rewrite rule that looks like it turns something true into something false.
A system is called sound only if every inference rule preserves truth under its intended interpretation.
A rewrite rule is harmless unless you’ve already given it a strictly defined meaning.


Right now, the notation


(B)-&gt;-( )-&gt;-(B) &lt;&gt; (_)<br>

doesn’t have a defined semantics. Without a defined meaning, you can’t claim it’s true, false, or anything else. So it can’t break soundness.


You can’t declare something “unsound” before you’ve even said what it is supposed to mean.


2. “To make it sound we must specify that this only happens on a metalogical level.”​


That’s actually backwards.
Metalogical statements are never part of the axiom system. They describe the axiom system from the outside. If the rewrite rule only works at the meta-level, then it is not an axiom of the system, and so there is nothing to “fix” in the system’s soundness.


In other words:
If it only works as a meta-statement, then it’s not an inference rule, and therefore it cannot make the system unsound.


3. Introducing truth tables with “don’t cares” and “empty sets”​


A truth table only makes sense once you’ve defined how each symbol maps to true or false.
Right now the truth table is being written before the meaning of the symbols is defined. That reverses the normal order.


Also, in classical logic:


  • the empty set is not automatically “true,”
  • and “don’t care” entries are part of engineering logic simplifications, not proof theory.

So the table:


A B A-&gt;B<br>1 1 1<br>1 0 0<br>0 1 X<br>0 0 X<br>

does not correspond to any known implication. It’s simply redefining implication (“->”) into something that no longer behaves like implication. That’s allowed if you want to make a new connective, but then you must say:




Otherwise the discussion gets circular.

'

4. “Who’s to say it isn’t plausible?”​


Once the semantics are not fixed, anything is plausible.
A formal system doesn’t become sound or unsound because an expression “looks plausible”—it becomes sound only after you:


  1. give exact meanings to each symbol,
  2. define precise inference rules, and
  3. show that those rules never turn a true statement into a false one under that meaning.

Right now, the expressions he’s manipulating don’t yet have a defined semantics, so nothing can be proved or disproved about their soundness.
Click to expand...
Wat if you define meaning of (B)->-( as the same meaning as (B), i.e. a structure "B"
exists).

The empty structure is always "false". I changed A:AA to: (B)->-( )->-(B) <> (_) ->0, to add an indication that we reasoned through a logical singularity.
 
Just to mention that @talanum1 chose to bring religion(? or similar) into his last "science solution", which got it shut down as not being standard science.

No personal problem with that man, you should believe what works for you, for sure.


But otherwise done responding :thumbsup:
 
@talanum1 , are you aware that your Grey aliens are considered by many to be abusive clones? I hear there are some good aliens too, could have different hypotheses

Good luck.
 
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