Can a "real" paradox exist?

Question

Given a statement, S:

"S is not true."

We arrive at a paradoxical solution whether or not we assume S to be true or false. Does this automatically imply that we have made an error in logic, reasoning and/or the structure of the original statement? Alternatively could it imply that we are incorrect in assuming that true and false are exclusive?

So to extend on this thought, is it possible for a paradox to arise given that there are no errors in the reasoning and logic applied to the original problem?

Answer 1

I would try to answer this in terms of computation (nothing fancy)

Let's do some substitutions:

S = S is not true. #This says we can replace 'S' with 'S is not true'.

S = S is not true is not true.

S = S is not true is not true is not true.

We can carry on this process until there is no further substitutions to make. Well, that condition won't ever come in this case as this is plain old infinite recursion due to self reference. If you can't reach a condition where no further substitutions can't be made you simply cannot reach a conclusion. In this particular case you cannot say S is true or false.

Similarly if you email a person about some information and he never replied (in terms of whether he knows about it or not) for whatever reason, would you say that the person knows the information or he doesn't.

Does this automatically imply that we have made an error in logic, reasoning and/or the structure of the original statement?

I wouldn't call infinite recursion an error, it is more about practical purpose and infinite recursion is of no practical use to us mortals as we cannot use it to reach any conclusion. But in terms of computation an infinite recursion/loop is a bug :)

Answer 2

The first step in sorting out your question is to admit that a body of logic is made up of axioms and that there are multiple interesting potentially "correct" bodies of logic (I am unfamiliar with the jargon logicians use for "body of logic", formal system perhaps).

Next is to understand a theorem called the Principle of Explosion, which asserts that a contradiction in a body of logic implies that any statement is true. You can easily find proofs online for this statement. Such a body of logic, one where any statement is true, might be called the trivial one. It would be "valid" in the sense that it's a body of logic but invalid relative to the body of logic we like most; this last phrase being a loaded one.

So in summary, a paradox cannot exist in a given body of logic unless it is the trivial one. Since humans tend not to believe that every statement is true, we believe that there are no paradoxes in our reality.

Edit: I want to add the disclaimer that the Principle of Explosion might depend on some axiom that doesn't exist in some body of logic and that perhaps there is an interesting logic in which it doesn't hold. I hope someone might comment on this post to let me know.

Re-edit: Looks like there is a concept called paraconsistent logic which rejects the principle of explosion (either by removing the law of excluded middle it seems or some other way). So if you subscribe to a paraconsistent logic then you can have a universe in which paradoxes exist.

Answer 3

There are several paradoxes that already exist. The most well-known one would be the matter/energy paradox of light.

a) Light is a wave and not a particle. 
b) Light is a particle and not a wave.

Both statements are proven to be true.