Strange behavior in the liar paradox, Russell's paradox, Cantor's diagonal argument, and more problems

[tarmstrong]

Hello everyone,

I have found that a certain strange truth value, different from true and false, appears in all of Russell's paradox, the liar paradox, Cantor's diagonal argument, the Grelling-Nelson paradox, the crocodile dilemma, a problem related to the halting problem I came up with, and Berry's paradox. I am certainly not an expert in these problems, but I have not seen my arguments anywhere else. I think that my arguments might be new and that they say something important about these problems. I would really like to see what you think. My article is on my web page as "The Infinitely Recursive Truth Value" at http://www4.ncsu.edu/~tjarmst3/ .

I am hoping some people here will be willing to discuss it. I enrolled in a computer science Ph.D. program this fall, but there are not any professors at my university who will discuss this project or my Semantic Web project. So I am still on my own finding people with whom to discuss it.

I describe the truth value in my article like this:

 

 

Sometimes the only means of finding the truth value of a logical statement, considered in either a natural or formal language, includes finding its truth value as part of the process. For some statements, when we ask if they are true or false, by the nature of the statements we keep asking if they are true or false over and over, infinitely, and never arrive at true or false. As part of the procedure to find the truth value of a statement, we need to find the truth value of the same statement. We write a computer procedure to compute a predicate. When we call the procedure with certain arguments, inside the procedure is a recursive call to the same procedure with exactly the same arguments. The procedure to evaluate the predicate recurses infinitely when we use it to decide if a statement is true or false. To find the truth value of the statement, we have to keep trying to find the truth value of the statement again and again, forever. It is a truth value because some statements are true, other statements are false, and still other statements, when we ask if they are true or false, have this other behavior.

 

Do you think that's interesting? I have one formal and two informal arguments on the liar paradox. Here is my one informal argument:

 

 

Statements, whether in natural or formal languages, make claims about reality. In order to find the truth value of a statement, we have to ask if what it claims is correct or incorrect. If we want to find the truth value of the statement "2 + 2 = 4", we can find out if it actually does by running the addition function and the equality predicate. If we want to find the truth value of the statement "Paris is the capital of France", we need to find out somehow if the claim is correct.

So, we ask if "This statement is false" is true or false. It claims that it is false. The claim that the statement is making about reality is that the statement is false. We need to find out if its claim is correct. We need to find out if it is false. We need to ask if it is false. So we ask if the statement is true or false. Asking if the statement is true or false is what we were doing at the beginning. So we ask again. In order to find out if the statement is true or false, we have to find out if it is true or false. When we ask if the statement is true or false, we ask if it is true or false as part of the process. If it is false, it is true. We repeat the whole process of asking if the statement is true or false from the beginning. When we ask if the statement is true or false, we keep asking if it is true or false over and over again, forever. It is true if it is false. It is true if it holds that it is false. So we have to ask if it is false. "This statement is false" has the recursive truth value.

It is similar with "This statement is true." We ask if "This statement is true" is true or false. It claims that it is true. It is true if it is true. It is true if it holds that it is true. So we have to ask if it is true. In order to find out if it is true, we have to find out if it is true. Asking if it is true is what we were doing at the beginning. We ask if it is true or false again, and again and again, forever.

 

Do you think that's correct? Have you seen that before?

 

Then I wanted to show you my argument for Russell's paradox too. I couldn't get all the LaTeX to come out right in the forum, so I am attaching a PDF.

So, it is exactly the same behavior in both the liar paradox and Russell's paradox. When we ask if a statement is true or false, we keep asking if it is true or false over and over, forever.

Another problem in my article is Cantor's diagonal argument, but it is a bit long to explain in a post here. I came up with a certain way of formalizing the diagonal and anti-diagonal as sets in which the truth value appear and in which we actually can include the anti-diagonal as a row in the matrix. I also came up with something about the halting problem that I think is very interesting.

So, thank you so much if you have any comments!

Tim Armstrong
http://www4.ncsu.edu/~tjarmst3/

Russells_paradox.pdf

[studiot]

Is recursion a valid method of determining truth values?

 

How about the following program line:

 

100 If you have not reached this line end program.

 

Incidentally the liar paradox contains 2 statements rolled into 1 so you have the difficulty

 

Determine the truth value of

 

(2+2=4 and 2+2=3)

 

Clearly unresolvable, but quite resolvable if split into component statements.

[overtone]

 

 

I am certainly not an expert in these problems, but I have not seen my arguments anywhere else. I think that my arguments might be new and that they say something important about these problems.

Have you run across "Laws of Form" by G. Spencer-Brown?

 

He employs a convenient notation for the question, and describes an arithmetic of logic in which the truth value you have noticed is handled explicitly (a parallel with the arithmetic of imaginary nuimbers is noted).

[tarmstrong]

Thank you very much for your responses. I really appreciate it.

I have read some of "Laws of Form". I may not be remembering correctly, or I may not have gotten far enough into the book, but I don't remember him talking about infinite recursion with these problems or, when we ask if a statement is true or false, asking if it is true or false over and over again. Does he talk about that?

Oh, I'm sorry, studiot, I'm not quite sure what you mean. What do you mean by the liar paradox containing two statements?

[overtone]

 

 

I don't remember him talking about infinite recursion with these problems or, when we ask if a statement is true or false, asking if it is true or false over and over again. Does he talk about that?

Yes, he does.

 

It's something one runs into immediately when introducing variables into expressions of the form (an immediate nesting of expressions) or recognizing the dependence of the form on the topology of the space in which it is inscribed. He handles this by introducing time as a dimension, allowing him to continue to depict expressions on a page of a book. But I refer you to the book - worth reading, in its entirety.

[studiot]

 

Oh, I'm sorry, studiot, I'm not quite sure what you mean. What do you mean by the liar paradox containing two statements?

 

 

 

Well the original version ran along the lines of

 

Mick the Martian says all Martians are liars.

 

Which is a combination of the statements.

 

Mick is a Martian.

 

All Martians are liars.,

 

which may posses independent truth values.

[tarmstrong]

Thank you for the reference, then. I shall have to read "Laws of Form" in more detail.

So I'm clear, do you think my arguments for the liar paradox and Russell's paradox are correct? You are just saying people already know about them?

Edited October 31, 2013 by tarmstrong

[tarmstrong]

I chose the wrong focus for my post. What I would really like the most from posting on these forums and what would make me very happy would be if people could make progress on the halting problem based on what I have in my article. I found the truth value in a problem related to the halting problem, and it is also in Cantor's diagonal argument. I posted about the halting problem in the computer science forum at http://www.scienceforums.net/topic/79883-can-you-make-progress-on-the-halting-problem-based-on-this/ . Are people aware that this truth value is also in Cantor's diagonal argument, the Grelling-Nelson paradox, the crocodile dilemma, and Berry's paradox?

I ordered a copy of "Laws of Form" and am looking forward to reading it. Well, I don't know if people here thought my arguments for the liar paradox or Russell's paradox are correct in the first place. Some of my friends think they are correct. I just think that any time anyone discusses the liar paradox or Russell's paradox, they should give these solutions. "This statement is true", "This statement is false", and [math]P \in P[/math] are neither true nor false but instead have this truth value. It is just the behavior of these statements when we ask if they are true or false. I think these solutions should be in the Wikipedia articles on the liar paradox and Russell's paradox for one matter, but they are not there.

Thank you again for your responses. Yes, there is the version of the liar paradox with two statements, but I think we would say of the version "This statement is false" that it is just a single statement? Then I'm sorry, but I still don't know what you mean by this line: "100 If you have not reached this line end program." It is supposed to be a line in a computer program? It seems to me that if the program reaches this line, all that happens is that the "end program" statement is not executed, but I'm sure I am not understanding what you mean.

I was able to find the truth value in Russell's paradox because I first found it it two really strange sets I discovered:

 

[math]A = \{x \: | \: x \in A\}[/math]


[math]B = \{x \: | \: x \not\in B\}[/math]


They are the sets A and B. Something is an element of A if it is an element of A. Something is an element of B if it is not an element of B. B is very similar to the set in Russell's paradox. Have you seen these sets before? I think they are actually very funny. They are at the beginning of my article. I can explain them more here if you like.

[overtone]

What I would really like the most from posting on these forums and what would make me very happy would be if people could make progress on the halting problem based on what I have in my article. I found the truth value in a problem related to the halting problem, and it is also in Cantor's diagonal argument.

Sorry: I wasn't just posting the reference and then ignoring the thread, it's just that I have nothing to contribute on the topic of halting a computer program.

 

The reference to Laws of Form was intended as a starting point - the book is quite old now, and I have seen hints in various articles that at least somewhere people have greatly elaborated on it, and are employing its notation etc in logic circuit design among other fields, but I have no familiarity with any of that.

 

One issue, that may be relevant right now: you are not talking about one truth value, but a pair of them - analoguous to T/F, you have something one might call "+,- imaginary" as one refers to positive and negative "imaginary" numbers (the roots of unity come in pairs). In analogy to traversing the unit circle of the typically depicted complex plane, setting out counter clockwise or clockwise; in electrical engineering, the phase of the oscillation.

 

For a computer entering a loop, the question would be which "way" it is traversing the loop - in the example of the Liar's Paradox and those others there (except maybe the Cantor example, which i haven't followed) this would depend on whether one accepted the initial statement of the minor premise as T or F.

 

This approach will not "solve" the halting problem in the theoretical sense, btw. That is proven. Progress in actual programming is what one can hope for. My own intuition is that quantum computing will require this logical extension explicitly, for efficient coding, but really - I don't know.

Edited November 7, 2013 by overtone

[tarmstrong]

There are positive and negative sorts of infinite recursion in my article. I'm not sure if this is what you are thinking of, but you have given me a lot of ideas, so thank you very much! There are statements that are true if they are false and false if they are true, like "This statement is false". In Russell's paradox, [math]P \in P[/math] is true if it is false and false if it is true. There are other statements that are true if they are true and false if they are false, like "This statement is true". The infinite recursion looks different in these two types of statements. Most of the statements in my article fit these two forms, but the crocodile dilemma does not. I was originally making a distinction between them and having two versions of the truth value, but because of the crocodile dilemma I have just been having a single truth value. I will add some to my article to discuss the distinction now, though. Thanks for the ideas!

I think people generally regard statements that are true if they are false and false if they are true as contradictions. I am saying we want to rephrase the expression. We should say the statement "This statement is false" or [math]P \in P[/math] is true if it holds that it is false and false if it holds that it is true. Phrased this way, the expression is not a contradiction but instead infinite recursion. When we ask if the statement is true or false, we have to find out if it holds that the statement is true or false.

Now the statements just have the single truth value, the recursive truth value, instead of two different truth values at the same time. They are true if they are false and false if they are true. However, they are neither true nor false. They are not true, so we cannot conclude they are false. They are not false, so we cannot conclude they are true. They are not contradictions.

I think people usually don't regard "This statement is true" or the set of all sets that are members of themselves as problematic? They really are. "This statement is true" is true if it holds that it is true and false if it holds that it is false. So when we ask if it is true or false, we have to ask if it is true or false. I didn't include the set of all sets that are members of themselves in the PDF on Russell's paradox I attached to my first post, but it is in my article in the section on Russell's paradox:

 

Here is a variation on Russell's paradox:

[math]P^\prime = \{x \: | \: x \in x\}[/math]

[math]P^\prime \in P^\prime[/math] also has the recursive truth value. We ask if [math]P^\prime[/math] is an element of [math]P^\prime[/math]. It is an element of [math]P^\prime[/math] if it is an element of [math]P^\prime[/math]. It is an element of [math]P^\prime[/math] on the condition that it is an element of [math]P^\prime[/math]. So we have to ask if it is an element of [math]P^\prime[/math]. Asking if [math]P^\prime[/math] is an element of [math]P^\prime[/math] is what we were doing in the first place. So we ask again, and again and again, forever.

 

We ask if [math]P^\prime[/math] is a member of itself. It is a member of itself if it is a member of itself. So we have to ask if it is a member of itself. [math]P^\prime \in P^\prime[/math] is true if it holds that it is true and false if it holds that it is false. Have you seen this argument for this set before?

If anyone here knows of any more statements than those in my article that are true if they are false and false if they are true, or the other way around, you might want to consider if they have this truth value. Let me know about it if you do know of any more.

Oh, thanks so much for talking with me. I really appreciate it.

[tarmstrong]

I read Laws of Form, and I don't see that my arguments are anywhere in the book at all.

The truth value provides a comprehensive solution to all the major paradoxes and to Cantor's diagonal argument, and it says something interesting about the halting problem. The reason I am convinced my arguments are new is that I have talked with a large number of professors at multiple universities about the truth value, or at least told them what the truth value is, and none of them have ever heard of the truth value in any of these problems. My problem now is just that I don't know how to get my results public.

[PeterJ]

Laws of Form is a solution for Russell's Paradox. Brown does not talk about your problem because it is not really relevant to his calculus, (for which such recursions do not occur). But somewhere Brown does talk about metaphysical problems and discusses the way they make us oscillate like an old-fashioned electric bell back and forth between two polarities. He introduces into ordinary logic a special value in order to solve the problem. He does not deal with your problem ('This statement is false') because (I think) he does not see R's paradox as taking this form.

[overtone]

 

 

Laws of Form is a solution for Russell's Paradox. Brown does not talk about your problem because it is not really relevant to his calculus, (for which such recursions do not occur).

Recursions of exactly the kind referenced in the OP (the Liar's Paradox etc - first level self-reference generally) , yielding a truth value explicitly described as a resolution of Russell's Paradox and with obvious relevance to similar situations (as far as I can tell, although I haven't tracked it through Cantor's original proof, it's exactly the truth value described in the OP) are discussed in the book Laws of Form.

 

An analogy is drawn to the arithmetic of negative numbers, and those familiar can extrapolate to the roots of unity generally.

 

A chapter is devoted to the matter of simple notation, with Brown pointing out that he should "really be writing in three dimensions" and introducing recursion in time as a means of handling self reference in circuitry (and vice versa). An application of such recursion/self reference to a real physical problem - a self-referential logic circuit capable of counting without a clock - is pictured in that book.

 

It is dated, last generation's progress, but I confess to being puzzled by the repeated claims of irrelevance.

[PeterJ]

I bow to your expertise. I'd just want to award it a better mark than 'last generation's progress'. This generation has yet to catch up with Brown's metaphysics. But yes, on the maths I'm weak.