Trivialism is Trivially True

Everything is true, and nothing is.

Today, I'm going to show you how to prove trivialism - the claim that all propositions are true. This may seem very counterintuitive at first, but you'll come out of this essay knowing the undeniable truth. All of the undeniable truths, actually.

Proof 1: Set Theory

Let P be some arbitrary proposition, and consider the set C which contains all sets X such that, if X is a member of itself, P is true. In set-builder notation,

\(C = \{ X\ |\ X\in X\implies P \},\)

where ⇒ represents material implication.

Now, we ask the question: Does C contain itself? Suppose that it does. Then, since C is a member of C, it must be the case that, if C contains itself, P is true. This is because C is defined as the set of all sets with this property. But C does contain itself, so P is true.

Thus, by assuming that C contains itself, we were able to derive P. So, by conditional proof, C∈C ⇒ P. But this is just the condition for C to be a member of C. Therefore, C does contain itself, and by modus ponens, P is true.

Since this proof holds for an arbitrary proposition P, it follows that all propositions are true.

Proof 2: From Dialetheism

Dialetheism is the position that there are true contradictions. It is of course an entailment of trivialism, since trivialism says that for any proposition P, both P and its negation are true, thus producing a true contradiction. But it’s long been known by logicians that the existence of even one true contradiction implies trivialism. This is known as the principle of explosion and can be proven as follows:

\(\begin{align} 1.\ & P \land \lnot P \\ 2.\ & P & \text{from 1, conjunction elimination}\\ 3.\ & P \lor Q & \text{from 2, disjunction introduction}\\ 4.\ & \lnot P& \text{from 1, conjunction elimination}\\ 5.\ & Q & \text{from 3 and 4, disjunctive syllogism} \end{align}\)

Here, P and Q are arbitrary propositions, and ∧, ∨, and ¬ mean “and,” “or”, and “not,” respectively. As we can see, by assuming any contradiction, we can prove an arbitrary proposition, and thus, dialetheism entails that all propositions are true.

So all we need to do to prove trivialism is to find one true contradiction. But there are many examples of this. For example, sometimes people use the same word to mean multiple different things. My house might be big by one standard of bigness but not by another. Thus, my house is both big and not big, a contradiction!

Dialetheism can also be shown to be true by logical paradoxes. For instance, consider the barbershop paradox: Imagine that there’s a barber who shaves all the people in town that don’t shave themselves. However, this barber is Sweeny Todd, and he murders all his customers, so all the people who don’t shave themselves are dead. But you, dear reader, are a person who reads articles about philosophy and logic on the Internet. Clearly, you have your head too far in the clouds of abstract ideas to perform basic practical tasks like shaving yourself. Thus, you must be one of Sweeny Todd’s customers. So how are you still alive to read this?

Now, you might still think that dialetheism is false, and I agree with you! After all, I’m a trivialist, and “Dialetheism is false,” is a proposition, so it must be true. But this poses no problem for dialetheism - it just implies that dialetheism is itself a dialetheia.

Proof 3: The Arithmetic Proof

Propositional logic can be turned into arithmetic by representing truth values as 0 and 1, with 0 representing false, and 1 representing true. The various logical operations can then be expressed as mathematical functions - for example, negation is expressed by the function f(x) = 1-x, and conjunction is represented by the minimum function or by multiplication. Now we know from the law of the excluded middle that every proposition is either true or false - in other words, it has a Boolean truth value of either 0 or 1. So what if we could prove that 0 = 1? Well, that would mean that all propositions have a truth value of 1, i.e., that all propositions are true.

So how can we prove that 0 = 1? Let’s start with Grandi’s series:

\(G= \sum_{n=0}^\infty (-1)^n = 1-1+1-1+1-1+\cdots\)

We can evaluate the sum with some simple arithmetic:

\(\begin{align} 1-G &= 1-(1-1+1-1+\cdots) \\ &= 1-1+1-1+1-\cdots \\ &= G\\ 1&=2G \\ G&= 1/2 \end{align}\)

Now consider the alternating sum of all natural numbers:

\(A= \sum_{n=1}^\infty (-1)^{n-1}n = 1-2+3-4+5-6+\cdots\)

What happens if we add it to itself?

\(\begin{align} 2A&=A+A \\ &= 1-2+3-4+5-6+\cdots \\ &\ \ \ \ \ \ \ \, + 1-2+3-4+5-\cdots \\ &=1-1+1-1+1-1+\cdots = G\\ \therefore A&=\frac12G = \frac14. \end{align}\)

Now we can consider the regular sum of all natural numbers:

\(N= \sum_{n=1}^\infty n = 1+2+3+4+5+6+\cdots\)

Surely this one has to just be infinity, right? Well (yes because trivialism is true, but) actually…

\(\begin{align} -3N&= N-4N \\ &= 1+2+3+4+5+6+\cdots\\ &\ \ \ \ - (\ \, 4 \ \ \ \, + \ \ \ \, 8\ \ \ + \ \ \ 12 \ \ \ +\cdots)\\ &= 1-2+3-4+5-6+\cdots=A=1/4.\\ \therefore N&=-\frac1{12}. \end{align}\)

Now we’re almost there. We have one more sum to consider:

\(S= \sum_{n=1}^\infty 1 =1+1+1+1+1+1+\cdots\)

Notice that S is just equal to N – N:

\(\begin{align} N-N &= 1+2+3+4+5+6+\cdots\\ &\ \ \ \ \ -( \,1+2+3+4+5+\cdots)\\ &= 1+1+1+1+1+1+\cdots \end{align}\)

Thus, S = 0, since we showed that N has the finite value -1/12, so S = (-1/12) – (-1/12). Now we are ready for the final proof:

\(\begin{align} 1 &= 1+0\\ &= 1+S \\ &= 1+(1+1+1+1+1+\cdots)\\ &= 1+1+1+1+1+1+\cdots\\ &=S=0.\\ \therefore 1&=0. \end{align}\)

1 = 0, so truth = falsehood. QED.

Proof 4: The Ontological Argument for Trivialism

Consider the most convincing possible proof that 1+1=3. Obviously, a proof that actually exists would be much more convincing than one that doesn’t exist - how are we supposed to convince anyone with a proof that doesn’t exist? Therefore, the most convincing possible proof that 1+1=3 must actually exist. But if there actually exists a proof that 1+1=3, then 1+1=3, since it has been demonstrated by the proof. We can run the same argument with any proposition, thus proving that all propositions are true.

Interestingly, I think that this argument itself may be the most convincing proof, thus demonstrating its own existence.

Proof 5: The Modal Ontological Argument for Trivialism

Consider the following argument:

Premise 1: It is possible that trivialism is true.

Premise 2: Necessarily, if trivialism is true, then it is necessary that trivialism is true.

Lemma 1: Therefore, it is possibly necessary that trivialism is true.

Conclusion: Therefore, trivialism is true.

This argument is valid in S5 modal logic, so the conclusion is true as long as the premises are. But Premise 2 follows from the definition of trivialism: By definition, if trivialism is true, then trivialism is necessarily true, since, “Trivialism is necessarily true,” is a proposition, and all propositions are true under trivialism. Since any proposition that is true by definition is logically necessary, and logical necessity entails metaphysical necessity, Premise 2 is true.

The only controversial part of the argument is Premise 1, but surely non-trivialists should not be so dogmatic as to suppose that it’s not even possible that they’re wrong. What hubris! Also, I can imagine a possible world where God tells me that trivialism is true, and since God is infallible, that would make trivialism true in this world.

Now of course, there is a reverse ontological argument showing that trivialism is false, but this is not a problem - see the Objections section.

Interestingly, renowned logician Kurt Gödel once created his own version of the modal ontological argument. He was supposedly arguing for God, but an automated theorem prover showed that his axioms were inconsistent, and thus he actually produced a proof of dialetheism and therefore trivialism.¹ I suspect that this was the contradiction he found that would allow the United States to be turned into a dictatorship, since it could now be proven from trivialism that the Constitution already says it is one.

Proof 6: The Equality of All Men

Speaking of the Constitution, as a proud American, I love all our founding documents, especially the Declaration of Independence. There’s just one problem with it, though - when Thomas Jefferson wrote, “We hold these truths to be self-evident,” he only followed it up with three truths rather than with every proposition expressible in the English language. Luckily, this problem can be rectified, as we can deduce trivialism from the first of his self-evident truths.

To do this, we will have to use Liebniz’s law of the indiscernibility of identicals. In symbolic logic, it says,

\(x=y \implies (\varphi(x)\iff\varphi(y))\)

where x and y represent any two individuals and φ can be any property. In plain English, it says that if x = y, any property of x is a property of y and vice-versa. Since the Declaration of Independence says that all men are equal,² this implies that all men have the same properties.

How can we deduce the truth of trivialism from this? Well, consider Danny Devito and Shaquille O’Neal. Both are men, so according to the self-evident truth, Devito = Shaq.

Two pictures of the same person

According to Wikipedia, Danny Devito is 1.52 m tall and Shaquille O’Neal is 2.16 m. The indiscernability of identicals thus implies that Shaq is 1.52 m tall as well, and Devito is 2.16 m. So, since Shaq and Devito are both 1.52 and 2.16 m tall at the same time, it follows that 1.52 = 2.16. Now we can subtract 1.52 from both sides to get that 0 = 0.64, and dividing by 0.64, we get 0 = 1. As was mentioned in the arithmetic proof, this implies trivialism.

For another proof, consider two actors (or rather, one actor): Dwayne Johnson and Daniel Radcliffe.

Also the same person

As we can see, The Rock’s dome has about as much hair as an actual rock, while Radcliffe’s head is as Harry as the Potter he played in those movies. From the indiscernability of identicals, then, we can conclude that each of them both has hair and doesn’t have hair. This is an instance of dialetheism, so it implies trivialism by the principle of explosion.

Incidentally, this gives us the answer to the old philosophical question of whether the present king of France is bald, or whether he has hair. The answer is both.

Proof 7: The Bayesian Argument

Bayes’s theorem says that

\(P(H|E) = \frac{P(H)P(E|H)}{P(E)}\)

where P(H) and P(E) are the prior probabilities of a hypothesis H and of making some observation E, P(E|H) is the probability that you would make the observation E conditional on H being true, and P(H|E) is the new probability of H after seeing the evidence. In other words, Bayes’s theorem says that, after seeing some evidence E, you should update your beliefs by multiplying your initial credence P(H) by the likelihood ratio P(H|E)/P(E). So, your credence should increase whenever the likelihood ratio is greater than one and decrease when it is less than one.

So how does this prove trivialism? Well, any piece of evidence you could possibly observe is logically entailed by trivialism, since trivialism says that all propositions are true. So P(E|trivialism) = 1 for any E. This means that the likelihood ratio will always be at least one, since P(E) is between 0 and 1. But it will almost always be less than one, since most things you observe are not things that you were certain³ of beforehand. In fact, for any useful evidence, P(E) < 1: If P(E) = 1, then observing it wouldn’t give you any evidence for or against hypotheses that didn’t already have zero credence. So any time you go around observing stuff, you are constantly getting evidence for trivialism! Everything you’ve ever seen, heard, etc., is evidence for trivialism. In fact, it’s often quite strong evidence - after all, the prior probability of observing the exact thing that you observed should be very low, given non-trivialism, but given trivialism, it’s 100%. And your own existence is anthropic evidence for trivialism, too. Even if you started out with a very low credence in trvialism, the probability after all these updates should be incredibly high due to the sheer quantity of observations and the giant likelihood ratios involved.

Proof 8: POTUS

Donald Trump was recently inaugurated as the 47th president of the United States. But did you know that there have only been 45 presidents in total, including him? It’s true - look at this list of all the presidents. If you’re reading this in 2025, there are only 45 in total. So Trump is the 47th out of only 45 presidents. That implies that 47 is less than or equal to 45, since you can’t be part of the list of presidents if you’re after the last one. However, it’s also true that 45 < 47. Thus, by the transitive property of the less-than relation, 47 is less than itself. Perhaps it’s a recipreversexcluson?

So how does this prove trivialism? Well, it allows us to perform another arithmetic proof. Let’s start with the equation

\(\frac{47-47}{47-47} = \frac{47-47}{47-47}\)

This is of course true, since the left- and right-hand sides are the same expression. But wait, aren’t we dividing by zero? Well, don’t worry: Since 47 < 47, it follows that 47 – 47 > 0 (it’s also less than 0). Since it’s possible to divide by any positive number (or negative number), the expression on both sides is valid and well-defined. Now let’s evaluate both sides. On the left, we can first evaluate the numerator to get zero. Since zero divided by anything is zero, we find that the left-hand side is zero. On the right, we notice that the numerator and denominator are the same, so we can cancel them out to get 1. This proves once more that 0 = 1, so trivialism is true.

But this rabbit hole goes even deeper. In addition to being the 47th president, Trump is also said to be the 45th. So 45 = 47. Likewise, Grover Cleveland is said to have been the 22nd and the 24th president, so 22 = 24. These facts can likewise be used to prove that 0 = 1 and thus imply trivialism.

Proof 9: It's trivial.

It’s in the name, duh. Obviously.

Proof 10: From Trivialism

The final argument for trivialism goes like this:

Premise 1: All propositions are true.

Premise 2: Trivialism is a proposition.

Conclusion: Trivialism is true.

You might object that this is circular reasoning. After all, isn’t Premise 1 just the statement of trivialism itself? Well consider the following proof:

Premise 1: Circular reasoning is always sound.

Conclusion: Circular reasoning is always sound.

I rest my case.

Objections

Now that you’ve heard my arguments, you probably have some objections. That’s okay, and I totally agree. All your objections are right! After all, presumably all of your objections are propositions stating that there’s a flaw in one of the arguments, and thus, they are all true. You might also have some arguments against trivialism itself. I agree with those arguments as well! After all, trivialism is false. That’s one of the entailments of trivialism. It’s just that trivialism also happens to be true.

Pirate Captain explains whether trivialism is true.

1. Rather appropriate the the man who has both a completeness theorem and an incompleteness theorem named after him.

2. You might be concerned that it only says they are created equal and not that they are still equal. However, we can deduce from the fact that all men are created equal that all men are still equal, for if x=y at any time, then all properties of x and y must be identical at that time. But this includes properties about what x and y will be like in the future - for example, “x will be 6 feet tall in 2025.” And even if you reject this, we can run the same argument and just consider the properties of all men when they were first born - just replace height and baldness in my examples with birth weight and whether or not a baby has brown eyes.

3. or almost certain