Difficulty: What the fuck

I’m not going to teach you how to do logic in this post. That would take many hour-long lessons just to cover the tip of a dick-shaped iceberg called “logic.”

Instead, I want you to be able to recognize different types of logic, the same way you can recognize German, Spanish, Arabic, Mandarin, and Korean—even if you don’t actually speak any of them.

Most people hear the word “logic” and immediately check out. Their brain goes:

“Hey, did you know there’s a sneaker sale this weekend?”

But fuck that. Logic is about clear thinking, and clear thinking is basically a superpower.

So let’s at least figure out what the different “dialects” of logic even are. We will briefly discuss the gist of:

1. Classical Logic
2. Modal Logic
3. Intuitionistic Logic
4. Relevance Logic
5. Paraconsistent Logic

and bestow honorary mention on fuzzy logic at the end.

1. Classical Logic (The Default Setting)

This is the one most people unknowingly use.

It has two big rules:

(1) Big Rule 1: No contradictions allowed

You cannot have:

“Pringles are good”
and
“Pringles are not good”

both true at the same time, in the same sense.

This is called the Law of Non-Contradiction.

(2) Big Rule 2: No middle ground

A statement is either:

• true, or
• false

No in-between. No “kinda true.”

This is called the Law of Excluded Middle.

The really crazy rule: explosion

If you do allow a contradiction:

P and not-P

then everything becomes provable.

Yes—literally anything.

Example of Explosion

Start with:

It is raining AND it is not raining

From this, we can get:

It is raining
It is not raining

So far, so normal.

Now here’s the key move:

From:

It is raining

you can say:

It is raining OR the moon is made of cheese

(This is called addition—and yeah, it’s weird but legit.)

Now combine that with:

It is not raining

And use a rule (disjunctive syllogism):

If:

• A or B
• not A

Then:

• B

So:

• “It is raining OR the moon is made of cheese”
• “It is not raining”

Therefore:

The moon is made of cheese

Boom. You just “proved” nonsense.

Moral of the story:

Contradictions break the system.

This is called:

• Ex Falso Quodlibet
• aka Explosion

One-liner:

Classical logic = “No contradictions, no gray area, or everything goes to shit.”

Now we level up.

Instead of just asking:

“Is this true?”

we ask:

“Does this HAVE to be true?”
“Could this be true?”

2. Modal Logic (Must / Might Logic)

While classical logic talks about things like what is actually true (like the fact that many Singaporean drivers are assholes), modal logic talks about things like:

• You must wear a seatbelt.
• Some Singaporean drivers might be assholes.
• Some car accidents could happen

Classical logic is like:

“Cool story, but I only understand true/false.”

Modal logic is like:

“Let’s handle that shit properly.”

The Two Main Modal Words

□ (box) = MUST / necessarily

◇ (diamond) = POSSIBLE / maybe

Here’s a simple application to get you started:

□A = “A has to be true in all the situations we’re considering”

◇A = “A could be true in at least one situation”

And here’s an example:

“The toilet must be clogged.”

This means: in every possible situation we’re considering, the toilet must be clogged.

Here’s another one:

“The toilet might be clogged.”

This means: there is at least one possible situation where the toilet is clogged.

The key idea: “possible worlds”

This is the weird part, but we’ll keep it simple.

Modal logic imagines:

different ways reality could be

Not sci-fi necessarily. Just:

• what might be true
• what could have been true
• what has to be true

Think of:

different “versions of reality”

Like:

• World 1: it’s raining
• World 2: it’s sunny
• World 3: it’s cloudy

Now define the operators

◇A (possible A)

There is at least ONE world where A is true

□A (necessary A)

In EVERY world, A is true

Example

Statement:

“2 + 2 = 4”

• □(2+2=4) → true (it’s true in all worlds)
• ◇(2+2=4) → also true

Here’s another statement:

“It is raining”

then:

• □(raining) means: it is raining in all accessible worlds
• ◇(raining) means: it is raining in at least one accessible world

But suppose from the current world you can access three worlds:

• World 1: raining
• World 2: not raining
• World 3: not raining

Then:

• □(raining) is false, because not all accessible worlds have rain
• ◇(raining) is true, because at least one accessible world has rain

And if you want to be a tad nerdier, here’s how we symbolize this raining stuff:

□(raining) = it’s raining everywhere

¬□(raining) = not everywhere is raining

◇(raining) = somewhere it is raining

□¬(raining) = nowhere is it raining

3. Intuitionistic Logic (a.k.a. “prove it or shut up” logic)

Core Attitude

“Don’t tell me something is true unless you can actually prove it.”

What They Hate

In normal logic, you can say:

“Either A is true OR A is false”

Even if you have zero evidence.

Intuitionists say:

“Nope. That’s lazy as hell.”

In kiddy language, we can say:

Normal logic:

“Either there’s treasure in that box or there isn’t.”

Intuitionistic logic:

“Have you opened the fucking box?”

If not:

“Then don’t claim shit.”

What changes?

In classical logic:

You can prove things using tricks like:

“If assuming NOT-A leads to nonsense, then A must be true”

This is called proof by contradiction.

In intuitionistic logic:

“No. That doesn’t count.”

They say:

“You didn’t prove A.
You just showed that not-A is bad. That’s not the same thing.”

Simple example

Classical logic says:

“There exists a number that has property X”

even if you can’t name it.

Intuitionistic logic says:

“Show me the number or shut the fuck up.”

One-line summary

Intuitionistic logic =
“No proof? No truth. Stop bullshitting.”

4. Relevance Logic (a.k.a. “don’t say random shit” logic)

Core attitude:

“The reason (A) should actually be related to the conclusion (B).”

What they hate

In classical logic, this is allowed:

From a false statement, you can prove ANYTHING.

Example:

• “2+2=5”
• therefore: “I am the King of Mars”

This, again, is called explosion.

Relevance logicians say:

“What the actual fuck? These are unrelated.”

Weird conditional (“if…then…” statement) problem: kiddy version

Normal logic:

“If unicorns exist, then I’m a sandwich”
This is true if unicorns don’t exist.

Relevance logic:

“Dude… unicorns and sandwiches have nothing to do with each other.
This is bullshit. Reject.”

What changes?

Relevance logic demands:

A must actually have something to do with B.

You can’t just glue random sentences together with “if.”

Example

Classical logic allows:

“If 2+2=5, then the sky is blue.”

Relevance logic says:

“Nope. That’s not a real conditional. That’s just nonsense dressed up.”

One-line summary

Relevance logic =
“Your premise better actually connect to your conclusion, you dumbass.”

Why this pissed people off

Some philosophers said:

“Wait… real life systems have contradictions all the time.”

Examples:

• laws that conflict
• people who believe inconsistent things
• messy databases
• paradoxes

And they thought:

“Why should ONE contradiction destroy EVERYTHING?”

---

5. Paraconsistent Logic

Core idea:

“Even if there’s a contradiction, don’t let everything go to shit.”

What they reject

They reject this rule:

From A and not-A, you can prove anything

They say:

“No. That’s way too extreme.”

Kiddy version

Normal logic:

“If your notebook has ONE contradiction, we burn the whole fucking notebook.”

Paraconsistent logic:

“Relax. One bad page doesn’t mean the whole notebook is garbage.”

Concrete example

Suppose your system says:

• “This website is safe”
• “This website is not safe”

Classical logic says:

“Cool, now I can prove the website is run by aliens.”

Total nonsense is allowed.

What they are trying to do

They are trying to:

contain the damage

Instead of:

letting one contradiction explode into total chaos

Another kiddy version

Classical logic:

One rotten apple → burn the whole farm

Paraconsistent logic:

One rotten apple → throw it out, keep eating

Important: they are NOT saying

“Contradictions are good”

They are saying:

“Contradictions shouldn’t destroy everything”

Why this actually matters

Because in real life:

• legal systems contradict themselves
• large databases have errors
• people hold inconsistent beliefs

If we used classical logic strictly:

everything would become meaningless instantly

So paraconsistent logic says:

“Let’s build a system that can survive inconsistency.”

One-line Summary

Paraconsistent logic =
“Even if shit contradicts, don’t let the whole system lose its mind.”

Final comparison (all five now)

Logic Type	Attitude
Classical	“Contradiction = everything explodes”
Modal	Add “necessity and possibility”
Intuitionistic	“No proof = no truth”
Relevance	“No connection = bullshit”
Paraconsistent	“Contradiction ≠ total disaster”

Conclusion and a brief note on fuzzy logic

These are five major types of philosophical logics. (And there are more… many more.) Some of these logics extend classical logic. Others reject parts of it entirely. In the future, we will cover them in more depth as well as briefly discuss fuzzy logic, which, contrary to popular belief, is not a synonym for “sloppy thinking.” (It’s a specialty that math and engineering people love and has legit philosophical relevance… just not center stage.)

That’s all for now!