argumentation – Philosophy without Bullshit

What the Heck Is an Enthymeme? And Why Are They So Sneaky?

Difficulty: What the heck

Consider the following argument:

“I will be a little late today; traffic is bad.”

Formulated more clearly, the argument looks like this:

Premise 1: Traffic is bad.

Conclusion: Therefore, I will be late.

That looks fine at first glance, but there is a missing premise that we would need to make explicit for the argument to deductively support the conclusion. In other words, the argument looks more like this:

Premise 1: Traffic is bad.

Premise 2: ?

Conclusion: Therefore, I will be late.

So what is the missing premise? People often guess things like, “I’m driving,” or “I’ll be stuck in traffic.” Those are reasonable guesses, but they are still not enough to make the argument deductively valid. To do that, we would need a hidden premise like this:

Premise 1: Traffic is bad.

Premise 2: If traffic is bad, I will be late.

Conclusion: Therefore, I will be late.

Now the argument has the form of modus ponens, perhaps the most basic rule in logic. Briefly, modus ponens is:

P
P $\rightarrow$ Q
∴ Q

where ‘P’ = ‘Traffic is bad’, ‘P $\rightarrow$ Q’ = ‘If traffic is bad, I will be late’, and ‘∴ Q’ = ‘Therefore, I will be late.’

Without the second premise, the argument is deductively invalid.

Arguments like this one, containing at least one hidden premise or unstated conclusion, are called enthymemes.

Now, not all enthymemes are sneaky in a bad way. We leave premises unstated all the time because ordinary conversation would be unbearably tedious if we said every single thing out loud. Enthymemes become sneaky when the missing premise is exactly the part of the argument that is weakest, most controversial, or most in need of scrutiny.

That is why they matter so much.

If you fail to identify premise 2 (“If traffic is bad, I will be late”), then you have one fewer premise to target in the argument. You may find yourself trying to disprove premise 1 (“Traffic is bad”), which, if the traffic actually is bad, would make it much harder to criticize what is really going on.

But once we identify the hidden second premise, we begin to understand what to target: is premise 2 true? Is it really the case that if traffic is bad, then the speaker will necessarily be late?

Once we begin to question that previously hidden premise, we can come up with counterexamples. We can say: “If traffic is bad, you can still arrive on time.” Why? Because:

you can take the metro
you can work remotely
you may already have built in extra travel time
you may live close enough to walk

This is an easy example, but things start to get trickier once we get into advertisements and political arguments.

Consider this fictional advertisement that I made up just now:

Do you like being pampered? Do you want to be attractive, alluring, charming, sexy, whatever it is that floats your boat? Then you must apply our proprietary Bullshit Lotion.

First, let’s try to figure out what the argument is actually saying:

Premise 1: You like being pampered and you want to be attractive/alluring/charming/sexy/whatever.

Conclusion: Therefore, you must apply our proprietary Bullshit Lotion.

Once we do that, the missing second premise becomes obvious. The argument is really saying:

Premise 1: You like being pampered and you want to be attractive/alluring/charming/sexy/whatever.

Premise 2: If you like being pampered and you want to be attractive/alluring/charming/sexy/whatever, then you must apply our proprietary Bullshit Lotion.

Conclusion: Therefore, you must apply our proprietary Bullshit Lotion.

Now that we’ve identified the missing second premise, we can cast doubt on it.

Is it true that if you like being pampered and want to be attractive, etc., then you must apply the proprietary Bullshit Lotion? Perhaps not. Perhaps you can apply some other brand of lotion. Perhaps you do not need lotion at all. Or perhaps, if you want to be pampered and attractive, you can just go to a spa. You get the point.

Or take, for example, political enthymemes. Suppose that a politician says:

“My opponent wants to cut the military budget. So my opponent is weak on national security.”

What is the missing premise? And how might you cast doubt on that premise by providing plausible counterexamples?

I’ll leave that as a little exercise for you. But now you can see why enthymemes are so sneaky: what is doing the most argumentative work is often the very thing left unsaid.

What the Hell “Begging the Question” Really Means

Difficulty: What the hell

So it’s been around 18 years since my first girlfriend dumped me, but while I have long gotten over it, I have not come to terms with the stupidity of her argumentative strategy. Consider the first part of her argument:

“What your father does is wrong. Why? Because I feel really strongly about it.”

I asked her why she feels “really strong about it.” She responded:

“Because it’s wrong.”

This circular reasoning—this is what philosophers call begging the question—went on for a good fifteen or thirty minutes.

Today, we’re looking at the informal fallacy of begging the question. You just read an obvious example. Now, we’re going to look at some subtler instances.

Christopher W. Tindale has us consider, in his Fallacies and Argument Appraisal, the following arguments:

Example 1:

A heavier-than-air craft could never fly because in order to lift up and travel over distance a machine would have to be lighter than the environs surrounding it.

Example 2:

God is the only perfect being and perfection includes all the virtues. So, we know that God is benevolent.

He then asks two critical questions to identify instances of begging the question:

Critical question 1:

Has the arguer avoided the obligation to provide independent support for a claim by restating it in similar terms?

Critical question 2:

Has an arguer avoided the obligation to provide independent support by assuming somewhere in the premises the very thing that has to be shown?

I’m going to add one more question to the list:

Critical question 3:

Do the premise(s) and conclusion depend on each other for justification?

These questions will help us articulate how Example 1, Example 2, and the ex-girlfriend example beg the question.

First, there is no real question when someone begs the question

Let’s disabuse ourselves of a potential source of misunderstanding. Nobody’s really asking a question when they commit the “begging the question” fallacy. Here, the word “question” is better understood as the “issue at hand.” If, for instance, a Bible thumper tells you,

“God exists because the Bible says so, and the Bible is the word of God”

it’s not that anyone is asking a question. It’s that the arguer presumes the issue at hand—namely, whether God exists—to be true, when that is exactly what she needs, but fails, to independently support.

In other words, the premise “the Bible is the word of God” hides the assumption that God exists. So duh, if she cheats like that, of course she can (fallaciously) reach the conclusion that God exists. She’s basically saying:

God exists because God exists.

Restating a claim in similar terms

Here’s critical question 1 again:

Has the arguer avoided the obligation to provide independent support for a claim by restating it in similar terms?

Now revisit Example 1:

A heavier-than-air craft could never fly because in order to lift up and travel over distance a machine would have to be lighter than the environs surrounding it.

This one’s sneaky. But yes—the arguer has avoided the obligation to provide independent support by restating the claim in different terms.

Briefly:

“Heavier-than-air craft” ≈ “machine” that is not “lighter than the surrounding air”

“Fly” ≈ “lift up and travel over distance”

These expressions are not strictly identical, but they function the same way in the argument.

So the arguer hasn’t explained anything. She’s just saying:

A heavier-than-air craft cannot fly because a heavier-than-air craft cannot fly.

Sneaking the conclusion into the premises

Reconsider this argument:

God is the only perfect being and perfection includes all the virtues. So, we know that God is benevolent.

Sounds good, right? Good my ass.

Rephrased, the argument goes like this:

Premise 1: God is the only perfect being.

Premise 2: A perfect being has all the virtues (including benevolence).

Conclusion: God is benevolent.

This looks like an argument, but it isn’t doing any real work. The conclusion is already built into the premises.

So while it doesn’t look like “G, therefore G,” that’s basically what’s going on under the hood.

Mutually dependent premise and conclusion

Now back to the ex-girlfriend argument:

I feel really strongly about what your father does. Therefore, what your father does is wrong.

What your father does is wrong. Therefore, I feel really strongly about it.

If this were an Escher drawing, it’d be a masterpiece.

Recall critical question 3:

Do the premise(s) and conclusion depend on each other for justification?

Here, the answer is yes.

Each claim is being used to justify the other. That’s classic circular reasoning.

You might be tempted to ask whether circular reasoning and begging the question are really the same thing. Fair question—but we don’t need to settle that here. For now, it’s enough to see how these arguments go wrong.

A psychological note on begging the question

In A System of Logic, John Stuart Mill observes that people are unlikely to commit this fallacy in their own private reasoning, but they do commit it in dialogue.

My guess is this: people know their reasons are weak, but instead of admitting it, they just restate their conclusion in slightly different words and hope no one notices.

Don’t do that.

What the Heck Is the Fallacy of Denying the Antecedent? That Descartes Joke…

Difficulty: What the heck

René Descartes walks into a bar and orders a drink. When he finishes his drink, the bartender asks him if he would like another. Descartes replies, “No, I think not,” and disappears in a puff of logic.

Philosophers are probably not amused… not because the joke talks crap about Descartes (it doesn’t), but because its attempt at humor rests on a misunderstanding of basic logic. In fact, there are two fallacies at play here: equivocation and denying the antecedent.

Equivocation

Equivocation is a fallacy where a word or phrase has two different meanings that are mistakenly lumped together as one. This is an informal—not formal—fallacy. That is, the fallacy arises not from the logical form of the argument, but from the content of the argument. Take this meme:

The Bible says being gay is fine, as long as you’re high.

Why? Because…

“A man who lays with another man should be stoned.”

–Leviticus 20:13 ESV

The fallacy hinges on a (deliberate) equivocation on the word “stoned,” which can mean two things: (1) to be high and (2) to be executed by stoning.

Applied in a deductive argument, equivocation might look like this:

Premise 1: Saint Stephen was stoned.

Premise 2: If you’re stoned, you’re allowed to be gay.

Conclusion: Therefore, Saint Stephen was allowed to be gay.

Here, the form of the argument is what we call modus ponens, a deductively valid move in logic. But the content of the argument—specifically, the two distinct meanings of “stoned” in the premises—is conflated, creating an informal fallacy.

Equivocation in the Descartes joke

So where does equivocation fit in the Descartes joke?

“I think not.”

This has two different meanings:

“I think not”¹ = “I don’t think so”
“I think not”² = “I am not thinking”

The joke exploits this ambiguity, going from “I think not” in the first sense to “I think not” in the second.

Denying the antecedent

This is where it gets pretty stupid. Besides equivocation, the fallacy of denying the antecedent—a formal fallacy—must be applied for the joke to work at all.

Briefly:

Denying the antecedent:

P → Q

¬P

∴ ¬Q

This says:

If P, then Q

Not P

Therefore, not Q

This is invalid because Q could still be true for some other reason. But it’s a tempting mistake. Some instances of denying the antecedent look convincing at first glance:

Premise 1: If I die, I’ll be in hell.

Premise 2: I’m not dead.

Therefore, I’m not in hell.

But the fallacy becomes more obvious once we change the content:

Premise 1: If that’s a turtle, then that’s a reptile.

Premise 2: That’s not a turtle.

Therefore, that’s not a reptile.

Clearly, an animal can be a reptile without being a turtle.

In the case of the Descartes joke, the implied argument is:

If I think, I exist.

I think not (that is, I am not thinking).

Therefore, I don’t exist.

Same form, different content.

So there you go—a seemingly funny joke ruined by a brief logical analysis, which I find funnier than the joke itself.

And they say that Descartes “disappears in a puff of logic”?

I think not.

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).