What the Hell Is a Formal Fallacy? This Is Where Things Blur

Difficulty: What the hell

What the hell is a fallacy?

Philosophers draw a distinction between formal fallacies and informal fallacies.

Formal fallacies

Roughly, a formally fallacious argument is a deductively invalid argument where the form of the argument is wonky. Take, for example:

Premise 1: If I have COVID, then I display symptoms X.

Premise 2: I display symptoms X.

Conclusion: Therefore, I have COVID.

The form of this (bad) argument is:

C → D

D

Therefore C

Note that the conditional (if…then) is symbolized by an arrow going from left to right. That means C gets you D. But that doesn’t mean D always gets you C.

For instance, if it’s true that having COVID implies that I display symptoms X, I can’t conclude from the fact that I display symptoms X that I have COVID—because after all, I might have some other non-COVID lung problem that’s causing those symptoms.

And sure, it might also turn out that I do have COVID after all, but that possibility doesn’t make the argument any less fallacious. The problem is not that the conclusion happens to be false. The problem is that the form of the argument allows for a case where the premises are true and the conclusion is false. That is what makes it deductively bad. So:

fallacy ≠ false

formal fallacy = a deductive argument whose form does not guarantee the conclusion

Here’s another example:

Premise 1: If that’s a dog, then that’s a mammal.

Premise 2: That’s a mammal.

Conclusion: Therefore, that’s a dog.

The form here is identical to the (bad) argument about COVID. We have:

D → M

M

Therefore D

But this one is very obviously fallacious.

Again, it’s fallacious not because the conclusion happens to be false (it might very well turn out that the mammal is a dog), but because even if the premises are all true, you still can’t say for certain that the conclusion is true.

Informal fallacies

So where do informal fallacies come in?

Take something that is, in classical logic, both necessarily true and formally correct:

A = A

A

therefore A

And say that someone asks you this:

“Who is the current president of the United States?”

You respond with this argument:

John Adams is the current president of the United States. Why? Because John Adams is the current president of the United States.

That’s stupid—not because there’s anything wrong with the form of the argument, which is simply A, therefore A, but because the argument is rhetorically useless and question-begging. It gives the hearer no independent reason whatsoever to accept the claim.

So the John Adams argument is stupid due to something that logical form alone cannot capture, namely, some conversational, rhetorical, epistemic (knowledge), or other informal aspect of language itself.

Or take this example, taken from an old DirecTV commercial:

“When you pay too much for cable, you feel down. When you feel down, you stay in bed. When you stay in bed, they give your job to someone new. When they give your job to someone new, he has a lot to learn. When he has a lot to learn, mistakes are made. And when mistakes are made, you get body-slammed by a lowland gorilla. Don’t get body-slammed by a lowland gorilla. Get rid of cable, and upgrade to DirecTV.”

The chain part of the argument has a deductively valid form. It’s basically:

A → B

B → C

C → D

D → E

E → F

F → G

So:

A → G

Now, to get from there to “therefore don’t pay too much for cable,” you need at least one additional premise—something like this:

¬G

Therefore ¬A

And even if you generously grant the commercial that extra anti-gorilla premise, what’s still fallacious is not the logical form. The form is fine. What’s fallacious is the evidentiary stuff: the more links you have in a chain argument, the more evidence you need to justify each link. Is there enough evidence to say that “when mistakes are made, you’ll get body-slammed by a lowland gorilla”? No. And the same goes for several of the other links.

Where the line blurs

An interesting issue arises when we consider borderline cases of fallacies. Take this one:

Premise 1: Some dogs bark.

Premise 2: Guai Guai is a dog.

Conclusion: Guai Guai (probably) barks.

It’s unclear whether this is best treated as formally or informally fallacious.

The case for “formally fallacious”

If we ignore the word “probably” and interpret this as a deductive argument, the argument is deductively invalid because the form does not guarantee the conclusion.

You can illustrate that with a neat little Venn diagram like this one.

The red X in the overlap of the dog-circle and barker-circle stands for premise 1: some dogs bark.

But that X only tells you that at least one dog is in the barking region. It does not tell you which dog.

Premise 2 tells you that Guai Guai belongs somewhere inside the dog-circle. But it does not force Guai Guai into the overlap. Guai Guai could be in the barking part of the dog-circle, or in the non-barking part of the dog-circle.

That is the crux of the problem: even if the premises are true, the conclusion “Guai Guai barks” is still not guaranteed to be true.

If you’re not familiar with predicate logic, ignore the next few lines. But if you do know it, you’ll see the point immediately:

∃x(Dx ∧ Bx)

D(g)

Conclusion: B(g)

That inference is not deductively valid. The existential premise says that some dog barks. It does not say that this particular dog, namely Guai Guai, barks.

The case for “informally fallacious”

Now this is where things get messy. If we interpret the Guai Guai argument not as deductive (an argument that attempts to conclude that something is logically guaranteed to be true if the premises are all true) but as inductive (an argument that attempts to conclude that something is more likely true than not—in short, a probabilistic argument), then the defect seems to be evidentiary rather than purely formal.

Revisit the argument:

Premise 1: Some dogs bark.

Premise 2: Guai Guai is a dog.

Conclusion: Guai Guai probably barks.

If “some” turns out to mean “just one dog” (which it might, because “some” means at least one), then the conclusion that Guai Guai probably barks is supported by very weak evidence.

On the other hand, if “some” turns out to mean “99% of dogs” (which is also consistent with the word “some”), then that would be enough evidence to support the conclusion that Guai Guai probably barks.

This suggests that once we explicitly add the word “probably,” the argument is no longer best criticized as a deductive screw-up. Instead, the problem is that the evidence may be too weak to support the probabilistic conclusion. That makes the defect look informal rather than formal.

So what?

The problem with this borderline case is that intro-level logic teaching often makes the formal/informal divide sound cleaner than it really is.

Intro logic books often focus on a small handful of named formal fallacies, such as:

affirming the consequent (the COVID argument form)
denying the antecedent
undistributed middle

Possibly that’s because such fallacies are common, psychologically tempting, and canonically named.

But I don’t think fallacies have to be common or psychologically tempting to count as fallacies; they just have to be logically or linguistically defective. And that whole “canonically named” thing? That starts to smell a bit like dogma.

And even if broadening the term “formal fallacy” ends up making that category messier than philosophers would like, so what? If a counterexample makes the old definition unstable, that doesn’t show that the counterexample is bad. It might just show that philosophers’ understanding of the formal/informal divide is wonky and needs revision.

That’s my spiel for today. Keep thinking (and arguing).

What the Hell Is an Argument? (And Why “Deductive = General to Specific” Is Bullshit)

Difficulty: What the hell

People love saying:

“Deductive arguments go from general to specific.”
“Inductive arguments go from specific to general.”

This is repeated so often that even professors say it.

Unfortunately, it’s wrong.

And if I hear one more person say that, I’m gonna have a fucking seizure.

First: what an argument actually is

An argument = a set of statements, a.k.a. premises, that lead to a conclusion.

So this is an argument:

Premise 1: If I hear one more person say “deductive arguments go from general to specific,” I’m gonna have a fucking seizure.

Premise 2: I hear one more person say “deductive arguments go from general to specific.”

Conclusion: Therefore, I’m gonna have a fucking seizure.

The form, or structure, of the argument is:

H → S
H
Therefore, S

That logical move, by the way, is called modus ponens. It’s a valid and “duh” move, yes, but the Latin makes it sound deep, and it’s exactly the kind of structure all deductive logic is built on.

In logic, an argument is NOT defined as:

yelling
a claim (like “postmodern art is valuable, but postmodernism is not”–sounds deep, but that’s just a claim/statement)
an opinion (including the ones everyone and their uncle feel entitled to express)
disagreement

What deduction REALLY is

Deductive validity

Deduction means:

If the premises (of an argument) are true, then the conclusion must be true.

In other words:

The truth of the premises guarantee the truth of the conclusion.

So, if ‘H → S’ (premise 1) is true, and if H (premise 2) is true, then S (conclusion) must be true.

Specifically, that’s what we call a deductively valid argument.

Notice I kept on italicizing the word if. That’s because that’s super important–after all, one or more of these premises might not be true. For example, is premise 2 (“If I hear one more person say “deductive arguments go from general to specific,” I’m gonna have a fucking seizure”) true? No. So deductive validity doesn’t mean that the conclusion of your argument is true. It just means, again, that

if the premises are all true, then the conclusion must be true.

Deductive soundness

But what if you have a valid argument and its premises are all true? For example:

Premise 1: If I don’t have any money, I can’t pay the mortgage. (TRUE)

Premise 2: I don’t have any money. (TRUE)

Conclusion: I can’t pay the mortgage. (BOTH LOGICALLY AND IN REALITY GUARANTEED TO BE TRUE)

Then this deductive argument is what is called sound.

A word on that bullshit definition of “deduction”

Note that in neither the seizure example nor the mortgage example is anything “going from general to specific.” That’s because “going from general to specific” is simply not the correct definition of “deduction,” nor is it useful for a deeper understanding of how arguments work. But some teachers and professors will still use that definition no matter how many times I object. Whatever. Let’s move on.

What induction REALLY is

Inductive logic does not use labels like “valid” or “sound.” Instead, we use words like “weak,” strong,” and “apt.” This different terminology is important because, unlike deduction, induction is probabilistic.

Inductive weakness

Consider this argument:

Premise 1: Some dogs bark.

Premise 2: Guai Guai is a dog.

Conclusion: Guai Guai (probably) barks.

This is an inductively weak argument because of what the word “some” means.

“Some dogs” might be just 1% of dogs. “Some dogs” might even be just 0.01% of dogs. There simply isn’t enough reason for us to accept the conclusion that Guai Guai probably barks because, by definition, probably = roughly put, “more likely than not” or “high enough likelihood given the evidence.” And the word “some” is simply too weak to conclude that Guai Guai’s barking is probable.

Inductive strength

Now what about this one?

Premise 1: Most Martians like Dr. Seuss.

Premise 2: Zorp is a Martian.

Conclusion: Zorp (probably) likes Dr. Seuss.

This is an inductively strong argument. “Most Martians” doesn’t mean some exact math like 51%. It just means a strong majority—enough to make the conclusion actually likely. So that–and the fact that Zorp is a Martian–makes it probable that Zorp likes Dr. Seuss.

Inductive aptness

What happens when you have an argument that is inductively strong and all its premises are true? That’s when you have an inductively apt argument. Let’s go back to the dog argument and change it a little:

Premise 1: Most dogs bark.

Premise 2: Guai Guai is a dog.

Conclusion: Guai Guai (probably) barks.

This dog argument has a form that is identical to that of the Martian argument, so it is obviously inductively strong. But unlike the Martian argument, the dog argument has premises that are all true. Hence, it is inductively apt.

A word on that bullshit definition of “induction”

Note that none of these inductive arguments “go from specific to general.” That’s because, like the bullshit-y definition of deduction, the bullshit-y definition of induction is not very helpful for a deep understanding of how arguments work. And yet, textbooks teach that definition. Perhaps I will be having that fucking seizure after all.

Another type of argument: abduction

There’s at least one other type of argument–the abductive argument–that we can discuss. Abduction, or inference to the best explanation (IBE), aims to reach conclusions based on the best possible explanation. Ockham’s Razor–the principle that the best explanation is the simplest one that makes the least number of assumptions–is a case in point. For instance:

Suppose you walk into your living room and see that:

The floor is wet
Your dog is shaking water everywhere
There’s a knocked-over bowl of water

You could come up with many explanations:

A pipe burst
Someone broke in and spilled water
Your dog knocked over the bowl

But one explanation stands out as the best:

Your dog knocked over the bowl and made a mess.

That’s an abductive argument:

The floor is wet and the bowl is knocked over.
If the dog knocked over the bowl, that would explain all this shit.
Therefore, the dog probably knocked over the bowl.

Notice what’s happening here:

You’re choosing the explanation that best fits the evidence.

A brief note on abduction and probability

Abduction is often treated as probabilistic, even if people don’t always say it that way.

When you say “this is the best explanation,” what you usually mean is something like:

This explanation makes the observed evidence more likely than the alternatives.

That’s exactly the kind of reasoning philosophers like Elliot Sober analyze using probability. (Check out his book, Ockham’s Razors.)

So if someone tells you that abduction is “not probabilistic,” they’re either oversimplifying or just wrong.

Conclusion

If you forget most of what I just said, remember this one thing:

The difference between deductive and inductive arguments has nothing to do with “general vs. specific.” It has to do with certainty vs. probability.

That’s all for today. And keep thinking.

Why You Shouldn’t Fucking Yawn When You Hear “Logic,” and Why There are Multiple Types of “Logics”

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:

Classical Logic
Modal Logic
Intuitionistic Logic
Relevance Logic
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:

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!