What in the Actual Fuck Is the Soundness Theorem in Metalogic? (Explained to Myself like I’m a Dumbass): Part 1

Before you attempt to understand this article, make sure you first read my previous article, “What the Fuck Does “Proof” Even Mean? (And Why It Doesn’t Guarantee Truth).”

I’m going to attempt the impossible in this post: explain something that confused me and pissed me off so much in grad school that it became one of the (many) reasons I quit the program. But it’s been a decade since I left, and now that there’s no pressure to publish papers, meet deadlines, and pass the damn courses, it’s time to revisit this shit with a vengeance.

Metalogic — what in the actual fuck does that even mean?

We’ll lay down some foundational metalogic concepts first so we can tackle what’s called the soundness theorem in metalogic within what’s called the Hilbert system, not just as a vague example, but with as much precision as possible—while translating every piece of symbol garbage into normal human English.

The calculator analogy

Say you have a calculator.

2 + 2

and it spits out:

4

Cool. That’s just doing math.

That’s like doing a normal logic problem: you follow rules → you get an answer.

But metalogic asks something way more annoying:

“How do you know the calculator won’t randomly give you 5 or 10 or 17 one day?”

Not just for 2 + 2.

But for:

• 17 + 83
• 999 + 1
• ANY numbers you plug in

So we’re not checking answers.

We’re checking whether the system itself ever screws up.

What we need to show

To prove the system is reliable, we need four things:

1. Premises
2. Axioms
3. Rule Case
4. Inductive Step

I’ll explain what the hell these things mean using analogies along the way.

(1) Starting assumptions (premises)

These are just things you’re told to start with.

Math analogy:

You get a math problem. It says assume that x = 5.

You don’t question the math problem and say shit like, “How do you know that x = 5? Can you prove it? No? Then shut the fuck up.”

Instead, you say, “Cool, let’s just say that x = 5 is true.”

Chess analogy:

Say that you’re playing a game of chess. You’re given a starting position: the white queen is on one square, the black queen is on another, the rooks are in the corners, and so on. You don’t say, “How do we know that’s where these pieces begin?” You just assume those positions are correct and move on.

Logic example:

Statement 1: If you eat Taco Bell, you’ll have diarrhea.

Statement 2: You eat Taco Bell.

…you don’t question whether these two statements are true. You assume they’re true for the sake of argument. If we want to add a few scary logic symbols, we write:

Γ = { Taco Bell, Taco Bell → diarrhea }

Translation (without the hieroglyphics)

Γ (the big scary Greek letter gamma) just means:

“the set of starting assumptions”

So this is just a fancy way of saying:

“Here are the things we’re starting with. This is like ‘assume x = 5’ in a math problem or a given starting position in a game of chess. Don’t question them (for now).”

Even simpler

Γ =

• Taco Bell
• If Taco Bell → diarrhea

That’s it.

No magic.

Important clarification (so nobody loses their mind)

We are NOT saying:

“Taco Bell is actually true in real life”

We are saying:

“Assume it’s true. Now let’s see what follows.”

One-line takeaway

Premises are just the starting point of the game—not guaranteed truth, just assumed truth.

(2) It starts with stuff that can’t be wrong (axioms)

Axioms are different from premises.

Premises are just assumptions you start with.

Axioms are patterns that come out true no matter what you plug into them.

Math analogy:

x = x.

No matter what number you plug in for x, it always comes out true. There is no situation where a thing is not equal to itself.

Chess analogy:

There are some things built into the chess system itself. One piece can’t occupy two squares at once. Two pieces can’t occupy the same square. If that happened, the game would already be broken before it even began.

Now here’s where my past self would start getting pissed off:

“Why the fuck are we allowed to just say this can’t be wrong? Aren’t we just assuming that?”

That sounds like what’s happening—but that’s not actually what’s going on. In logic and metalogic, it has something to do with a possible situation.

What the hell is a “possible situation”?

In logic, a “possible situation” does NOT mean:

• some weird sci-fi universe
• or “anything you can imagine”

It means something much more boring:

a way of assigning true or false to statements

That’s it.

That’s what truth tables are doing.

So we’re not assuming—it’s more like checking every possible truth assignment

Take a typical logical axiom:

“If A, then (if B, then A)”

Now we check ALL possible truth assignments:

A	B	B→A	A→(B→A)
T	T	T	T
T	F	T	T
F	T	F	T
F	F	T	T

This means:

• If A = true and B = true, then A→(B→A) is true.
• If A = true and B = false, A→(B→A) is still true.
• If A = false and B = true, A→(B→A) is true yet again.
• If A = false and B = false, A→(B→A) is true too.

We’ve exhausted all the possible situations, or all possible truth assignments. Whatever true/false shit you might assign A and B, A→(B→A) always comes out true.

In other words:

Does this thing EVER come out false?

No.

Not even once.

So we’ve proved that this is a legit axiom. That means we’ve proved that what we’re starting with can never spit out garbage.

Important clarification

You might say, “Hey, dumbass, what if I define a situation where A and ¬A are both true?”

For example:

A	¬A
T	T
F	F

Cool.

But then:

you’ve changed the rules of the game

You’re no longer doing classical logic.

And if you change the rules, then yes:

you have to redo the whole soundness proof for that new system

Final translation of “axiom”

Axioms aren’t trusted because we feel like it. They’re trusted because, given the system’s definition of truth, we can check every possible case—and they never come out false.

(3) The rules never mess things up (rule case)

Now we check the rules.

Math analogy:

Suppose you know:

• 2 + 2 = 4
• 4 + 4 = 8

Now you apply a rule:

“If a = b, then you can replace a with b”

So from:

• 2 + 2 = 4

you’re allowed to treat 4 as interchangeable with 2 + 2.

So (2 + 2) + 4 = 8.

This math rule never messes up.

Chess analogy:

Bishops move diagonally.

So if your bishop starts on a black square, it will always stay on black squares.

And if your bishop starts on a white square, it will always stay on white squares.

So the rule “bishops move diagonally” will never randomly make a bishop jump from a white square to a black square, or from a black square to a white square, or do some other crazy shit.

In logic, here’s the big rule:

Modus Ponens

A

If A → B

Therefore:

B

Concrete example

Metalogic is gay.

If metalogic is gay, Raymond’s ass is demolished.

Therefore:

Raymond’s ass is demolished.

Now here’s the important part

This isn’t about THAT specific sentence.

It’s about the form.

You can swap in anything:

Example 2

You eat Taco Bell.

If you eat Taco Bell, you’ll have diarrhea.

Therefore:

You’ll have diarrhea.

Example 3

I didn’t sleep.

If I didn’t sleep, I’m miserable.

Therefore:

I’m miserable.

Example 4

I make poop jokes.

If I make poop jokes, I’m childish.

Therefore:

I’m childish.

The key question

Is there ANY situation where:

A is true

“If A then B” is true

but B is false?

No.

There is no such situation.

Again, in intro to logic class, you can use a truth table to check this:

A	B	A → B
T	T	T
T	F	F
F	T	T
F	F	T

The only row where A is true and A → B is true is the first row. And in that row, B is true too.

Therefore:

This rule NEVER turns truth into bullshit.

(4) If every step is good, the whole thing is good (inductive step)

This is where grad school says:

“We proceed by induction”

and everyone (well, me, at least) dies inside.

Again, we’ll first use the math analogy.

Start with:

• 2 + 2 = 4
• 4 + 4 = 8
• 8 + 8 = 16

Notice something:

You’re always adding an even number to an even number.

And every time:

you get an even number

Now the key idea

If:

• even + even = even (this NEVER fails)

Then:

no matter how many times you repeat this process, you’ll ALWAYS get an even number.

That’s the inductive step

You’re not checking:

• just the first step
• just the second step

You’re saying:

“This pattern keeps working forever.”

And how do we know it works forever? We know not because we’ve checked a bunch of numbers and got tired and said, “Fuck it, it works for all numbers for infinity.” It works because we get the form of the math. That is,

2a + 2b = 2(a + b)

Steel staircase analogy:

You’re building an infinite staircase. The first step you build is made of solid steel. It doesn’t collapse. The second step is built the exact same way. It doesn’t collapse. The third step is also built identically to the second. So it’s impossible for the entire staircase to collapse.

Why?

Not because you checked a few steps and said, “Cool, it works for all the steps.” It’s because the way each step is built is identical to the next.

Chess analogy:

If every move you make is legal, and every move is made according to the same rules, then the entire game stays legal. The board won’t suddenly sprout a third king or summon a UFO to suck away your pieces halfway through the game.

Translation

If every step in your reasoning is safe, then the final result is safe.

Conclusion

If your eyes glazed over, don’t worry. Just keep this in mind:

Every line in a logical proof is either:

• a given assumption/premise (like x = 5)
• an axiom (like If A, then (if B, then A))
• the result of applying a rule (like modus ponens)

And induction is the method that lets us say:

if every step is safe, then the whole proof is safe

So if premises don’t start us with bullshit, axioms never spit out bullshit, and rules never turn truth into bullshit, then the whole logical system we’re checking is legit.

In other words:

If you can prove it (using the logical system you’re checking) (⊢), it’s true (⊨).

That’s all for today. In a future blog post, I’ll take you through the proof for the soundness theorem step by step, using the Hilbert system.