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.