# Can a “real” paradox exist?

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?

• First off, welcome to philosophy.se. This is a well-formed question. It's also a pretty common example in the philosophy of language called the "Liar Paradox". I'm actually surprised to see it has not been asked and answered here before. – virmaior Oct 30 '14 at 23:43
• This question is partially overlapping but might be a little hard to follow: philosophy.stackexchange.com/questions/6431/… – virmaior Oct 30 '14 at 23:47
• I feel like there's a very similar question here somewhere already (maybe asked by @MoziburUllah?) – Joseph Weissman Oct 31 '14 at 0:13
• The paradox you mention is a strengthened form of the original liar ('S is false'). The strenghtened liar does not even need the assumption that every sentence is either true or false; it only requires that sentences are either true or non-true. And it's conclusion is also stronger: S is both true and not true. There is a good collection of essays on variations of this paradox: Revenge of the Liar. Ed. Jc Beall. Oxford 2007. – sequitur Oct 31 '14 at 2:01
• There are different approach: some leave the assumption that true and false are exclusive : see Dialetheism. – Mauro ALLEGRANZA Oct 31 '14 at 8:14

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 :)

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.

• "Negation in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator." - en.wikipedia.org/wiki/Paraconsistent_logic Or, as CognisMantis pointed out with the light, just because you prove light is not JUST a particle OR a wave, does not mean that light is not a particle+extra stuff or a wave+ extra stuff or both a wave+particle or neither a wave nor a particle. – Nosajimiki Nov 15 '18 at 22:24

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.

• I guess there is an disjunction in between these 2 statements hence there is no paradox at all as you consider either one of the cases – Ankur Nov 5 '14 at 13:38
• If you put that in semantic logic and rearrange the statements a bit, you can easily see that they are contradictory, so they cannot both be true. But they are both true, so you have a paradox. – TwoThe Nov 5 '14 at 16:01
• but the statements aren't true. Light has characteristics waves and particles. It is more like this. A) Light has wave properties.B)Light has particle properties. – CognisMantis Nov 5 '14 at 18:35
• This is not a paradox. It is our lack of understanding and both theories describe a real physical phenomenon from different angles with great predictive power. No paradoxes in reality, just our incorrect assumptions – Madis Nõmme Jan 15 '19 at 14:56