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Well to be honest, my major is physics. And I only started some low level philosophy recently. So please criticize my opinions so that I can advance more quickly. :-)

My problem started with Russell's paradox. My philosophy teacher introduced Russell's paradox to me and gave some solutions such as type theory and ZFC. I noticed that these solutions approach the paradox by dealing with the definition of sets.

My question is that can we adopt a different solution by changing the definition of truth and false?

I assume that the logic value of a statement is not discrete but continuous. In other words, if I assign 0 to false and 1 to true like what has been traditionally done. Then in my new system I change this value to a real number between 0 and 1.

My basic idea is that every statement is essentially in the superposition of truth and false. Truth and false are like two wave functions, and together they form a new wave function. And that new function is the real state of a statement. Every time we check a statement, we retrieve a certain value (0 or 1), just like every time we make an observation the wave function collapses and what we get is actually one of the component of the 'big wave function'.

Then if I make the assumption above, I can know the approximate value of a statement by checking it for certain times. Every time I check I get either 0 or 1, and then I can know the value of the whole statement by calculating the mathematical expectation of my process. And this expectation is of course between 0 and 1.

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    What you were talking about is Fuzzy logic – Luis Masuelli Nov 6 '15 at 4:00
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So you have a very good idea, but it's not quite what the mathematicians were looking for as they dealt with the paradoxes of that age.

The big issue is that you are offering an approximation to a paradox, rather than actually resolving it. This is useful, mind you, but it's not what the mathematicians were looking for. In particular, they needed to make sure that the meanings of True and False withstood literally infinite applications without becoming ill defined. Redefining truthhood from a binary value to a real is meaningful, but its not the spirit those like Russel were going for. For one thing, probability is a COMPLICATED layer of mathematics, and they were trying to pin down the absolute simplest layers with the fewest assumptions possible, or even no assumptions at all. Starting from probability would be a tough pill to swallow.

The second issue is one that may appear as you flesh the idea out: there may be some paradoxes which result in infinitely divergent results, where categorizing the result with a bunch of draws averaging at 0.5 true does not capture the fact that the underlying math was ill defined.

Those issues aside, what you describe is not unlike a Quantum Computer. We've actually built them, and had them do things classical computers bounded by True and False, cannot match (well, fine... we've factored 15 into 3 and 5, but we did it in polynomial time, so chew on that, classical computers!) Quantum Computers solve some really neat problems that are quite unrealistic for classical computers to solve. Many even theorize that there exist problems quantum computers solve that can't be done with classical computers (part of the so called PSPACE class of computational difficulty)

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    QTM were shown very early in the life of Quantum Computers to not have any more power than regular TM in terms of computability. Their only advantage lay in their potential speed, but they were never supposed to solve undecidable problems or anything like that. – Alexander S King Nov 5 '15 at 5:46
  • @AlexanderSKing Very true. Perhaps I oversold them a bit. The part I was looking at was the portion of BQP which is believed to be in PSPACE, but not NP, indicating problems that are, as you say, solvable with a Turing machine, but not even a nondeterministic turing machine could do it in polynomial time. (alternatively, no deterministic turing machine could do it in exponential time) – Cort Ammon Nov 5 '15 at 5:50
  • I thought factoring problem could not be done in P yet... I thought Shor's algorithm was not yet ready... – Luis Masuelli Nov 6 '15 at 4:05
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Many values logics can very easily solves such problems. By stetting the truth value of the paradoxal statement to '1/2', thus by passing the law of non-contradiction. The issue then becomes how do you interpret this '1/2' value. In fuzzy logic (a many valued logic with continuous values), a process is used called defuzzification, where ultimately the membership functions are brought back to wither '0' or '1'. However I fail to conceive of a defuzzification process for your suggested problem. It seems to me that anything you gain from using many valued logic will be lost in the effort you put into to performing the correct defuzzification.

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You ask whether a solution of Russells paradox can be obtained by allowing truth values of propositions to range continously from zero to one.

It is not clear from your question which proposition you consider in the context of Russells paradox. More general, which statements do you have in mind for your proposal?

In the context of propositional logic you aim at replacing the truth function with values zero or one by a truth function psi with values in the intervall [0,1] of real numbers – or more in the spirit of quantum mechanics, in the domain of complex numbers of modulus <= 1.

In order to elaborate your proposal I would recommend:

  • Start with an example, which proposition do you want to consider?
  • Make clear what it means that the proposition has psi as truth function.
  • Make clear why the truth value collapses to the discrete values zero or one when checking the proposition.
  • Differentiate between the proposition, which is time independent, and our knowledge of its truth value, which is time dependent because it depends on the number of checks.
  • Make plausible how this proposal eliminates the Russell paradox.

Probably you may edit your question by clarifying these issues.

  • Sorry for the ambiguity. Russell's paradox here is not the main issue that I am considering. It just triggered my thoughts on the continuity of truth value. In addition, my understanding of both Russell's paradox and QTM are really shallow at this stage. I am just asking whether it is feasible or not to create a new logic system which is based on the continuous truth value. I'll be very glad if you can recommend me some papers or books on this topic. And given the fact that I am now preparing for an exam, I don't have enough time to mull over this topic. – Ezekiel Chen Nov 5 '15 at 8:58
  • @Ezekiel Chen Yes, fuzzy logic deals with truth values from the continous interval [0, 1] of real numbers. – Jo Wehler Nov 5 '15 at 16:09
  • But you cannot defuzzificate the truth value in this paradox – Luis Masuelli Nov 6 '15 at 4:09

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