# Algorithm for determining if a statement is a paradox

Please bear with me, since I have just started working on this.

Assuming that the laws of thought apply.

The basic goal of the algorithm is to recast the statement into true or false premises and check if a true and false clashes. If there is a clash then the statement or the set of statements is/contains a paradox. This is the rule for determining if a statement or a set of statements is/contains a paradox.

When I say clash, it means that a statement upon recasting cannot be said to have both true and false attributes.

Rule 1:

"This statement is true" is true. "This statement is false" is true.

The set of both statements is a paradoxical set.

Rule 2:

If in a statement there is a clash between true and false statements then it is a paradoxical statement.

"This statement is false" is true

"R, which is the set of all sets which are not members of themselves, is a member of itself" is True "R is a set of all sets which are not members of themselves" is true. "R is a set of all sets which are members of themselves" is false. "R is a member of itself" is true. Collectively it cannot be true, because there is a clash.

Now, we only need to recast the statement into true and false using the "not" operator to evaluate if a statement is true or false. We can evaluate all truth values and find if there is a clash between a true and false somewhere. If there is a clash then the collective set of statements or the statement is a paradox.

I got a couple in there. Can anyone throw me a paradox where this fails? I am trying to make it even simpler. Thank you so much! I think you will need some logic to recast more complicated statements, but this should identify the simple ones.

If nothing else, could you please verify this to be correct, and let me know if there is any procedure for finding if a set of statements contains a paradox currently exists. I think the general procedure detailing a clash between true and false in a paradox is absolutely correct.

I just came to know that there is a boolean procedure for finding self reference paradoxes, but it seems to me that my method might just be slightly more efficient, since it does not require solving for the entire set of statements but can halt whenever a clash is found. It could be hard to find if the set of statements has a solution or a clash or not, in the case of Wen's paper on the same, which recasts statements as boolean statements.

I can use this to disprove strong AI. Can someone please tell me if it is already proven that a human like intelligence cannot be recreated on a Turing machine.

So, there are a few problems with this.

• First, what is your formal definition of a paradox? Is it just a statement that can be found to be both true and false in some system? Paradox tends to be a bit of an informal term, so in CS we usually just refer to this as a contradiction, or an unsound logical system.

For example, Russel's paradox isn't actually a paradox, it's a proof by contradiction. We assume a premise i.e. "any set characterized by a logical proposition exists.", and show that we can prove falsehood from it, thus concluding that the premise must be false. (In intuitionist logic, this is actually how falsehood is defined).

• How are you encoding your logic statements? Traditional propositional logic doesn't give much in the way for describing self reference, so you probably need something higher order. Starting with natural language statements and converting them into logic is a whole separate problem.

• Your problem seems undecidable, since you could use it to determine if one can prove false from an arbitrary proposition, which means you can use it to determine if an arbitrary statement is true, which means you can use it to determine if a Turing Machine halts on given input.

• A) I need not define what a a paradox is, if there is a clash in True and False, I have found an inconsistency in the system, so you can say I am looking for inconsistencies. I am including premises and the set of statements which follow from that premise, if they can be derived. – user45490 Sep 10 '17 at 19:18
• B) This I have not done as of yet, I agree. This is informal logic being applied right now. – user45490 Sep 10 '17 at 19:19
• C) Can you give me an example? I am using informal logic here. Yes, this should be able to determine if an arbitrary statement is true using informal logic, assuming laws of thought apply. – user45490 Sep 10 '17 at 19:20
• Actually scratch that, there are no issues with the halting problem. This is informal logic, and this means that it would lose it's effectiveness when more assumptions are applied on top of it, like converting it into a mathematical format. But this isn't a mathematical concept, and one could use it to determine the truth of any arbitrary input. – user45490 Sep 10 '17 at 21:09
• So what exactly makes "this statement is false" a paradox, and "1 + 1 = 3" not a paradox? – gnasher729 Sep 10 '17 at 21:14

Since the precise logical framework to be used is not clear from the question, I'll use logic in a quite liberal way, as the generic "laws of thought" mentioned in the question.

I claim that the problem is at least as hard as deciding whether any formula $p$ is true. Therefore, it is undecidable.

For this, let $p$ be any logical formula, and define $q$ as the formula "this proposition is equivalent to $\lnot p$".

Since "this proposition is true" is not a paradox, while "this proposition is false" is one, it follows that $q$ is a paradox if and only if $p$ is true.

Hence, provability can reduced to paradox-checking.

• Can you give the complete logical process? You are making a few leaps here and there. I am sorry, but I have the precise logical framework written down, but it is too long to be put up here. – user45490 Sep 12 '17 at 17:15
• You are using formal logic, I think. It is definitely undecidable in formal logic, and hence cannot be coded in a Turing machine. I am giving an informal logic. I need only presume the sets, True, False and statement. Formal logic makes more assumptions. – user45490 Sep 12 '17 at 17:17
• You are also making things more complicated. Could you make them as simple as you can, without the need to make them more simple? – user45490 Sep 12 '17 at 17:22