The Bittersweet Paradox | On Mutual Exclusivity

Do you ever chuckle at people who see the face of Jesus on their toast? They find confirmation of their faith everywhere; it’s a knot in their wooden table, a message in the clouds, or a face at the bottom of their teacup. Rather than disparage them, we should try to understand their error: it’s a simple methodological mistake. They actively seek out “signs”, and therefore they find them everywhere.

This methodological error is not unique. I see it all the time, especially from irrationalists who argue that contradictions exist. They seek out paradoxes and think they find them everywhere. In fact, many irrationalists equate “enlightenment” with “coming to terms with contradictions”. The world is filled with so many contradictions, they say, that you have to be simple-minded to not see them.

If you’re familiar with my work, you know I am not sympathetic to irrationalism. The most common arguments come from appeals to quantum physics or the liar’s paradox. But I also frequently run into one argument that I’ve never seen properly addressed or labeled. So, I’ve given it a proper name: “the bittersweet paradox.”

The bittersweet paradox is yet another attempt to argue that contradictions exist. It goes like this:

We experience paradoxes and contradictions all the time. Take the simple feelings of happiness and sadness. They are contradictory emotions – happiness is the opposite of sadness, and sadness is the opposite of happiness – yet we often feel happy and sad at the same time. The opposites are unified into a paradox that we experience.

For example, say your mother just died after battling cancer for a year. She was in agony, night and day, but not anymore. What do you feel? Well, on the one hand, you feel awful because your mother died. On the other hand, you feel happy because at least she isn’t suffering anymore. So you feel happysad.

This phenomenon is just as contradictory as seeing the color “blackwhite” or Schroedinger’s cat being “deadalive”. But we clearly experience this feeling, so paradoxes must exist.

Put into a more abstract form, the argument looks like this:

  1.  P and Q are opposites.
  2.  Opposites are mutually-exclusive.
  3.  In situation X, P and Q are both true at the same time.
  4.  Therefore, contradictions exist.


There are a couple of errors nestled in this argument, both having to do with imprecise language. Specifically, we have to be clear what we mean by “opposites” and “mutually-exclusive”. The bittersweet paradox confuses the appearance of mutual exclusivity with actual mutual exclusivity.

Strictly speaking, mutual exclusivity is a logical relationship between two things. It means “there is no logical way X and Y can be true at the same time.” Consider the difference between these two examples:

  1.  I am happy right now.
  2.  I am sad right now.
  3.  I am happy and sad right now.


  1.  I have two legs right now.
  2.  I have no legs right now.
  3.  I have two legs and no legs right now.

In the first example, conclusion C is possible. I can be happy and sad right now, as explained earlier. “Happy” and “sad” are only colloquially seen as opposites – there’s nothing logically incompatible about them being unified together.

But in the second example, conclusion C is not possible. I cannot have “two legs and no legs” at the same time; that’s an actual contradiction. “Having two legs” is logically incompatible with “having no legs”.

This is precisely the error when people bring up bittersweet paradoxes. They simply highlight what appear to be mutually-exclusive phenomena – feeling happy and sad, cold and hot, or bitter and sweet. We don’t usually feel “cold and hot at the same time”, but that doesn’t mean they are mutually-exclusive.

It’s like entering a competition and getting in first place and last place at the same time. Have you experienced a paradox? Of course not; you could be the only one in the tournament. It doesn’t happen often, but it’s not any logical contradiction.

Consider one more example before moving on. Five hundred years ago, it would have seemed absolutely impossible to say the following: “I will be in Berlin on Sunday, and I will be in New York City on Sunday.”

But with modern technology, there’s no paradox or impossibility there. Anybody can fly from New York to Berlin in a day. This is why we need to be extremely careful in identifying mutually-exclusive relationships; what might appear impossible could be commonplace in a few centuries.

Not Knots

To avoid all possible confusion, we must dive into the weeds. Irrationalists flourish around murky language – whether they intend to or not – and we must join them to understand their errors.

We’ve established that the sentence “I am happy and sad at the same time” is not a contradiction. But watch what happens if we fiddle.

Instead of using the term “sad”, let’s just call it “not-happy”. Then, we can substitute one term for another, and we’re left with the following sentence:

I am happy and not-happy at the same time.

Now that sounds like a contradiction! If sadness is another term for not-happiness, then how do you get around this apparent paradox? You can imagine, in a very real way, that somebody could say “I am happy and not-happy at the same time” without contradicting themselves.

There are two errors at play here. First, and perhaps most obviously, the terms “sad” and “not-happy” are not identical. “Not-happy” is a negation – not something positive that you can be.

Which brings up the second error: an imprecise application of negation. In everyday language, putting a “not-” before a word turns it into a negation. But if we want to be precise, this won’t do, because it leads to ambiguity.

For any statement X, an improper negation is to say “not-X”.

A proper negation is to say, “It is not the case that X.”

So a logical contradiction is not “X and not-X”. Instead, it is “X, and it is not the case that X.”

That might seem pedantic, but it’s important. Consider our happy and sad example. The sentence:

“I am happy and I am not happy” could possibly not be a contradiction. It’s unclear what we mean by “not happy”.

But the sentence:

“I am happy and it is not the case that I am happy” is a logical contradiction. The first half of the propositions is properly negated by the second half – which means, by necessity, that if the first half is true, the second half is false.

Consider another example, to illustrate the importance of precise language. Say you are standing halfway inside a doorway.

Saying “Half of me is inside the doorway” is true.

Saying “Half of me is not inside the doorway” is also true.

Therefore, “Half of me is inside and half of me is not inside the doorway” is true.

You could see how an insistent irrationalist would try to turn this into a paradox. Proper negation clears everything up. An actual contradiction would look like:

“Half of me is inside the doorway and it is not the case that half of me is inside the doorway.”

If the first part of the sentence is true, then by logical necessity the second part must be false. No ambiguity present.

Quantum Paradoxes

Now, this might seem like elementary logic, but it’s applicable to many fields of thought.

Consider the ever-popular argument for paradoxes from quantum physics. As I’ve gone into detail about, people loudly and inaccurately proclaim that “quantum physics destroys classical logic”. It’s easy to understand why this is false.

According to standard theories in physics, “particles” and “waves” are mutually exclusive. Yet, in the double-slit experiment, light seems to exhibit properties of both waves and particles at the same time – and in really weird ways. So weird that people have concluded, “Reality itself is paradoxical! We have proof!”

Unfortunately for the irrationalists, this line of reasoning is flawed at a very basic level. Applying the simple principles I’ve discussed in this article resolves any apparent paradoxes.

First of all, to the extent any two things are experienced or are observed existing together, they demonstrate they are not mutually-exclusive. Obviously. What needs to change is the theories – no different than theories which insisted “it’s impossible to be in New York and Berlin on the same day.”

If people 500 years ago observed modern living, they wouldn’t conclude, “Wow! People can be in Berlin and New York on the same day. Reality is paradoxical!”

They would think, “Oh, I guess my theory about what was mutually-exclusive was wrong.”

The same is true in quantum physics. If we’ve observed some phenomenon which contradicts our theories, our theories are wrong; you aren’t witnessing a “true contradiction”. Perhaps “waves” and “particles” aren’t actually mutually exclusive. They might be two parts of the same coin – a “wavicle”, let’s say.

Carefully analyzing our concepts reveals a certain truth: whether X and Y exist at the same time is never a question of “whether mutually-exclusive things can be together.” It’s, “whether two things are mutually-exclusive.”

If two things are found together, they aren’t mutually-exclusive by definition, and it isn’t a hypothesis up for revision. To think otherwise is nonsensical, and it’s the central error in the bittersweet paradox.