Paradox Theory

[Wechtlein Uns]

I have oft heard the mention of "proof" as the inneffable source of mathematics, and , thus, of science. When I think about it, it makes sense. Science is a method of search for the truth, where the method of enquiry seeks to prove or disprove whatever statement you might think of about the world. Still, there is a twofold uneasiness between me and science, due to two rather strange observations:

1) it was Hume, I believe, who figured out, and proved mathematically, that there is nothing we can find in a cause that would lead us to deduce the resultant effect of that cause. This is troubling, because it calls into doubt the very validity of empericist thinking. Empiricism depends heavily on the idea that causes beget effects, and that the two are deductively linked. I am still considering this.

2) In two words, Kurt Godel. This is the man that also proved, mathematically, that there are an infinite number of theorems, statements one could make about the world that could not be proved in the negative, nor the positive. In this case, you should note that I am not talking about the statement "god exists", which can easily be proven given a proper definition of the term god. No, I am talking about contradiction.

Yes. Contradiction. If you were to write on a sheet of paper, "the statement on the other side of the paper is true", and then write on the other side, "the statement on the other side of this paper is false.", you would have produced two theorems about the world that could not be proven true nor false.

When Kurt Godel made this discovery, the entire mathematical world was in an uproar, because it effectively meant that a complete and internally consistent universe would require something beyond itself to solve these statements within itself. What a shock! I told this story to some religious friends of mine, and they immediately took it as a proof that God exists.

Indeed, the statement is unsettling. This thread is my response to Kurt Godel's discovery.

As I was pondering kurts argument, it struck me that all of these "incompleteness" theorems, so to speak, were always actually a pair of two or more concrete theorems. It strikes me that the statement, "the statement on the other side of this paper is true/false" depends on the statement on the other side of the paper for it's validity. What does the statement on the other side of the paper depend on for validity? Whatever it is talking about. So, the first observation I think is this:

1. Theorems make statements about other phenomena.
2. A theorem's validity is dependent upon the phenomena of which it speaks.

After some more thoughts, I realized that kurt godel's theorem relied upon the assumption that theorems are identical with phenomena. Think about it: Is a theorem, which speaks on phenomena, itself phenomena? I think we can agree that the defining characteristic of phenomena in general is existence, without which phenomena would not be phenomena. (it would be nothing). But a theorem on phenomena? Does it exist? Certainly, the concept of a theorem exists. Certainly, there can be little squiggles on a piece of paper that represent a theorem. But I am convinced that a statement about the world in general has no inherent existence.

Why? There was a clue in the first observation.  
To the ever expanding knowledge of the atheist mind!

[bowmore]

1) it was Hume, I believe, who figured out, and proved mathematically, that there is nothing we can find in a cause that would lead us to deduce the resultant effect of that cause. This is troubling, because it calls into doubt the very validity of empericist thinking. Empiricism depends heavily on the idea that causes beget effects, and that the two are deductively linked. I am still considering this.

It was indeed Hume.

2) In two words, Kurt Godel. This is the man that also proved, mathematically, that there are an infinite number of theorems, statements one could make about the world that could not be proved in the negative, nor the positive. In this case, you should note that I am not talking about the statement "god exists", which can easily be proven given a proper definition of the term god. No, I am talking about contradiction.

...<snipped>...

All of Kurt Godel's "paradoxical theorems" are a collection of theorems that depend on the results of all the other theorems and nothing else. Because of this, the theorems go in one big circle, forever trying to get to the bottom of definite phenomena, but only finding something that is dependent on something else for it's validity.

This is where you are mistaken, sadly. Not all Gödel statements are statements about statements.

Axiom : The barber cuts the hair of exactly those people in the village who do not cut their own hair.
Gödel statement : The barber cuts his own hair.

Sorry.

[Wechtlein Uns]

If you're saying that not all of godels statements were statements about statements, then I agree. I didn't think I was claiming that, but it could have sounded like that.

What I was saying is simply this: Statements that are recursive(i.e. depend upon, ultimately, the determination of their validity to be valid) are equal to "Null". They are not dependent upon phenomena for their validity. Add to that my argument that they have no inherent phenomenal existence of their own, and what do you think?

[bowmore]

If you're saying that not all of godels statements were statements about statements, then I agree. I didn't think I was claiming that, but it could have sounded like that.

What I was saying is simply this: Statements that are recursive(i.e. depend upon, ultimately, the determination of their validity to be valid) are equal to "Null". They are not dependent upon phenomena for their validity. Add to that my argument that they have no inherent phenomenal existence of their own, and what do you think?

But the example of a Gödel statement I gave, was not a recursive one either.

Another example.

Axiom 1 : It doesn't rain or I use an umbrella
Axiom 2 : I don't use an umbrella or it rains
Gödel statement 3 : It rains.

BTW your proposal that Gödel statements are equal to null, would require another logic as it is a contradiction of the axiom of excluded middle.

[Wechtlein Uns]

Oh, I think I understand what your getting at.

I believe there are many different types of statements. But of all the statements that might or might not be experienced, It seems that they all require something other than themselves for their own validity. A statement that says, "it rains", depends upon whether there is a phenomenon that can be singled out as "raining", in my view.

Now, as far as science goes, the material is all that exists. But it's slowly turning about that this material isn't really different from other material. Rather, all that there is--is phenomena. Pretty much. I understand that there are abstract concepts like respect or failure that might seem like they are independent of the phenomenal world, but I see it otherwise. In looking at phenomena, one does not have to slice the universe up in an instant. Phenomena changes, and one can group changes in phenomena as a phenomenon in its own right. I don't want to get into all the ways one can slice up phenomena, but suffice it to say, any statement that has any meaningfulness at all can ultimately be traced back to directly experienced phenomena.

I feel that, when someone makes a statement that is a paradox, it is because the statement relies not on phenomena for it's validity, but on another statement. Since a statement can not have any inherent validity in itself, the second statement must dependo n phenomena for the first statement to become proven true or false. But what if the second statement depends on the first statement for its validity? This, in my opinion, is what I consider to be the structure of a paradoxical argument.

Now then, the reason I say that the two statements, taken as whole, must be equivalent to null is because if they were dependent on something that was not null, they would be able to be proven true or false. If the two statement system depends not on phenomena, but on what then? All that there is is phenomena. The only other thing that is not phenomenal is null.

I agree that these kinds of statements can not be proven true or false. However, I do not think that that means some inherent flaw in the mathematical system. And that was my original point. Kurt Godel seemed to prove that there are mathematical statements that can not be proven true or false with a complete and internally consistent mathematical system. My main point is that there is not just True or false among statements. True or false is to be used when the statement depends upon some mathematical or phenomenal result to see its validity. Whereas null is the result of statements that ultimately depend upon null for their validity. and it seems that paradoxical statemnts rely not on phenomena, but "not-phenomena", thus what else could they rely on, but null?

I think, however, that you might have misunderstood my argument. I believe that there are statements that can be proven true and false, and there are statements that can not. But the statements that can not be proven can still be evaluated within an internally consistent and complete architecture of symbols.

Really, all I'm try to say is that there can be an evaluation of a paradoxical statement.

[Wechtlein Uns]

Axiom 1 : It doesn't rain or I use an umbrella
Axiom 2 : I don't use an umbrella or it rains
Gödel statement 3 : It rains.

.

At first glance I thought that this was a perfect example of what I was talking about, but It looks like this isn't. Followoing Godel statement three, it is obvious that "it rains", and "I use an umbrella" are chosen. What I would like to point out, however, is that this is not a paradoxical statement. The entire thing depends upon the phenomena of it raining. They don't depend on each other in a circle for their validity, and thus are not a paradox.

Soo...I don't see how this helps you at all. I all ready agreed that these types of statements exist, no?

[bowmore]

Axiom 1 : It doesn't rain or I use an umbrella
Axiom 2 : I don't use an umbrella or it rains
Gödel statement 3 : It rains.

.

At first glance I thought that this was a perfect example of what I was talking about, but It looks like this isn't. Followoing Godel statement three, it is obvious that "it rains", and "I use an umbrella" are chosen. What I would like to point out, however, is that this is not a paradoxical statement. The entire thing depends upon the phenomena of it raining. They don't depend on each other in a circle for their validity, and thus are not a paradox.

Soo...I don't see how this helps you at all. I all ready agreed that these types of statements exist, no?

Whoops I messed up axiom 2.  

Axiom 2 : I don't use an umbrella or it doesn't rain

That should fix it right? i.e. now there is a paradox.

My point was that the Gödel statement was not reflexive. It doesn't refer to itself or another statement.
But you are obviously just referring to the fact that within this system the truth of (3) implies it being false.

I'd like to explore your null value approach.
So please answer me these (to establish the grammar involved with nulls) :

not null = ?

true and null = ?
true or null = ?

false and null = ?
false or null = ?

[Wechtlein Uns]

Not null = ... if it's apparent, it's not null.

In the null valua approach, it's not enough to say that a statement can be either true or false, and then tack on null, as in "true and null". It doesn't work that way, but rather a statement is either true, false, or null. A true statement mirrors the phenomena it represents. While a false statement invokes phenomena that is, for whatever reason, not apparent. or rather, phenomena that doesn't match up with apparent phenomena. And finally, a null statement would be a statement that does not depend on phenomena at all for it's validity, but either another statement which in turn depends upon the previous statement, or a statement that depends upon itself for validity. I don't think it's possible to make as tatement that depends upon itself for validity, BUT, it is possible to have a chain of statements, of whatever number, that ultimately depend on each other for validity in one big circle.

So, in response to your questions, just as you can not have a "true and false", you can not have a "true and null". However, you can have a "true or null".

[Wechtlein Uns]

Whoops I messed up axiom 2.  :eek:

[bowmore]

Whoops I messed up axiom 2.  :eek:

The axioms aren't trying to reflect actual reality.

[bowmore]

So, in response to your questions, just as you can not have a "true and false", you can not have a "true and null". However, you can have a "true or null".

I'll try to clarify my question. Since you introduce a third possible value for 'booleans' the oprators in the algebra need to be able to cope with that new value.

the not() operator :

not(true) = false
not(false) = true
not(null) = ?

same for the 'and' and the 'or' operator.

Suppose a statement A is null (e.g. This statement is false)
And a statement B is not null (e.g. It rains)

What then is the value of the statement
not(A)
A and B
A or B

I wonder whether you are suggesting an alternate algebra to dodge Gödel's theorem. If so I'd like to explore that algebra to assess if you've been able to succesfully pull this off.
If you're just using null as a symbolic value to mark something as a Gödel statement, then this exploration isn't needed, but it'd also not be the great solution to Gödel's theorem.

[Wechtlein Uns]

I spent a lot of time intricately describing the system in detail, when all of a sudden, I press post, and lose the connection.  ), I'll save it for next time. Better yet, try and figure out why it is so! What's life without a challenge, eh?

I would also like to add that in the tripartite boolean system, there is also this:

(True or False or Null) = (True or False or Null) Null (True or False or Null)

If B is True, then:

(Null) Null (True) = True. Ah, that's the simplest one.
We'll I'll be leaving you to think about this, bowmore. I hope you come up with some interesting stuff. It's been a pleasure talking with you.

[McQ]

Excuse me for saying so, but......

holy shit!

 :pop:

[bowmore]

Or is like so:

(True or False) = (True or False) False (True or False). In the old algebra.
In the new: (True or False or Null) = (True or False or Null) False (True or False or Null)

It may be me, but I don't understand this notation, even in the old algebra.

What does (x)False(y) mean?

Therefore, if we are assuming that B is true(again, for simplicity), then:

(Null) False (True) Not(=) False, therefore (Null) False (True) = (Null) False[/i] (True) = True!

This was where I explained in detail why False correlates with or and True correlates with And, but after the lengthy description I allready gave(that was lost to a connection error

Well, I'm stuck until I can make sense of your syntax.
I'll play with it a little, perhaps it becomes clear.

[Wechtlein Uns]

Right. where to begin? I'm going to try to explain my syntax, though, if I'm being honest, I have yet to figure out the actual axiom for using true and false correlates with and (and) or. But I do have a very strong hunch that there's something solid there. I have a nose for sniffing out stuff like this.

First off, i think I need to clarify what we mean when we use the term "true" or "false". When a statement is true, all we're really saying is that the phenomena image evoked by that statement matches some phenomena somewhere in the universe. It's a matter of the Identity axiom, to say that Statement A = True. I believe that the And and Or operators operate in essentially the same way, except they are used with multiple statements to find one Truth or one Falsehood.

When I say: (True or False) = x, the True or false on the left are the possible answers to the statement on the right, namely, "X". The thing is, however, that "X" in this case, consists of more than one statement. So how are we to determine whether X is true or false? Just like with And and Or, we can require that all Statements be either true or false, or we can require that any statements be true or false. I would like to point out here, however, that when we say "all statements" we are, in essence, evoking a "True" type operator. Put it this way:

If A = B then (a and b) = True.

If A = True
And B = True,
the (A = B) = (True = True) = (True). What I've noticed here, is that in order for the statement to be True, Both statements have to inherently match up. And what do you call a statement that matches up? You call it True. That's why I equate "True" with And.

On the flip side, and this might be easier to explain, falsehood comes up because there is no match between a statement and percieved phenomena. The Or operator operates, on a basic level, upon the non-matching up of the multiple statements! If both A and B are true or false, then yes the statement is True or False accordingly, but we're actually calling upon an AND statement to do that. Whereas, if A is true and B is falses, then it is Or that does it's work. Or depends upon the two statements Truthfullnes and Falsity to not match up. And what do you call a statement that doesn't match up? You call it False. That is why I use False in my syntax in replace of or.

So: In the old algebra:

(true or False) is a basic or statement. It means the statement is either True OR false! However, it could be re-written as: (True) False (False) and still retain it's meaning.

The next true or false has to do with the truthhood or falsity of the first statement. It could also be re-written as (True) False (False). The truth is, "And" operators and "Or" operators can both be written in completely boolean logic. This has been done in Computers, where A string of Trues(1's) and Falses(0's) Represent And and Or. This is nothing new, and has been done in computer science before.

So:

(True or False) = (True or False) True (True or False) is simply the computer scientists way of representing an And statement with 0's and 1's. My above new operators, with (True or False or Null) uses a Tri-partite system. In this system, because there are three types of logic, there is actually an extra operator! I haven't named this operator, but it stems from the Null boolean.

anyways, I hope that clears things up. I'm not very good at putting my ideas forth, but if you still have questions, I should be able to answer them.