Unlocking the Secrets of Logical Fallacies: The Crucial Difference Between Affirming the Consequent and Undistributed Middle

Logic is a lot harder than people give it credit for, and logical fallacies are no different. Both are frequently misunderstood and misused on the internet, and it can often be us atheists/sceptics and other folks that claim to be logical that make the mistakes.

Sometimes there is the idea that because we know the fallacies, we don’t make fallacious arguments and that we call out fallacies correctly.

A question came up in a recent conversation about two fallacies that looked similar, and the level of detail I ended up having to go into with my explanation made the case for me that I should write this article.

What is a Fallacy?

A fallacy is an error in reasoning. It doesn’t necessarily mean the conclusion reached is false, just that the method to get to that conclusion is faulty. That said, fallacious reasoning often leads to false conclusions, so we should do our best to esnure our reasoning is as good as possible.

• “False Dichotomy!” – The Dilemma of Arrogance and Willful Ignorance
• Is That Really a Fallacy?
• Composition Error Fallacy
• Trolley Dave’s Fallacies [Video Playlist]

The Fallacies of Affirming the Consequent and Undistributed Middle

The fallacies of affirming the consequent and unditributed middle relate to arguments that, whilst not necessarily written in syllogistic form are best written that way as they make it easier to understand.

The Basic Syllogistic Form

When we are talking about the basic forms, we are talking about the structure of an argument. We will be looking at 2 types of syllogism, the first being a conditional the second being a set theory.

Examples and Explanation

With the logical forms, especially in short form, and other terminology around logic, it can sometimes be confusing or unclear.. so I shall do my best to explain all the elements.

A Sound Syllogism

In logic or, more precisely, deductive reasoning, an argument is sound if it is both valid in form and its premises are true, which in turn means the conclusion is true.

The Fallacy of Affirming The Consequent

To affirm the consequent is, to claim that the consequent is true. In committing the fallacy of affirming the consequent, one makes a conditional statement, affirms the consequent, and concludes that the antecedent is true.

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In Response to Daniel Goldman on Belief

We can see that there is a problem here because the consequent, or conclusion doesn’t entail the antecedent or initial conditional (premise).

If I eat 100 turnips at once, I will get a tummy ache.
I have a tummy ache.
I ate 100 turnips.

Whilst eating 100 turnips entails a tummy ache, having a tummy ache doesn’t mean 100 turnips have been eaten at once. In this instance the conclusion could be true, but isn’t necessarily true.

The Fallacy of The Undistributed middle

The undistributed middle is a type of logical fallacy that occurs when the middle term of a syllogism is not distributed in both premises. Rather than being a conditional statement with entailments, it is used in set theory.

We can be a little more specific, perhaps talking about items and qualities.

In this example, the conclusion is clearly false.

But what about if we rephrased the set theory syllogism as a conditional syllogism?

Rewording to Affirm the Consequent

Rewording a set theory syllogism into a conditional syllogism is possible, though sometimes cannot be done. Even when it can be done, you might have to alter the statement to add or remove elements to make it work.

If I eat turnips, I get a tummy ache
I have a tummy ache
I ate turnips.

They are similar, but their structure is different and the focus is different.

You could infer that perhaps the statement “if I eat turnips then I get a tummy ache” could mean that “all turnips give me a tummy ache.”
But then the next bit is different:

X – gave me a tummy ache
Therefore X must be a turnip

Is different to

I have a tummy ache
Therefore I must have eaten a turnip

All turnips give me a tummy ache is also different to the conditional statement if I eat a turnip I get a tummy ache.

The Bulldogs

They gave an example as to why they felt it was the same.

So, let’s apply this to the bulldogs are dogs example.

You end up with the same conclusion, which is why it seems like these mistakes in reasoning are the same.

As an example, we can look at math problems that end up with the same conclusions:

• 1+1 = 2
• 3-1 = 2
• √4 = 2
• 1*2 = 2
• 16/8 = 2

Just because these all ended up with the same answer doesn’t make them the same.

Conditional statements and Set Theory might sometimes end up with the same conclusion but they can be applied differently, there isn’t always overlap and it’s ignoring the purpose and structure of these two types of arguments.

Conditional vs Set Theory

The point of the if-then (conditional) is that it’s an argument about conditions (precedant, requirement, antecedent, causes etc) that need to be filled to bring about the result (entailments, conclusions, consequents, effects, entailments).

Whereas set theory is specifically referring to sets; groups of things, species, people or ideas that already, even necessarily, share a quality. To be part of that set you must have that quality.

When you commit the fallacy of affirming the consequent, you’re affirming the THEN statement, and saying the IF must be true. e.g. If the conclusion is true the premise must be true.
You could see it as including or leading to an either/or fallacy (aka false dichotomy), as either both are true or both are false.

If I swim then I get wet.
I am wet, therefore I swam.

The preblem here is being wet does not necessarily entail having gone for a swim. One could have a shower, walked through a sprinkler or been sprayed with a super-soaker, for example.

• What is a False Dichotomy? – Highlight – YouTube
• “False Dichotomy!” – The Dilemma of Arrogance and Willful Ignorance
• 7 False Dichotomy Fallacy – YouTube

When you suffer the undistributed middle it is saying that you haven’t distributed enough set information into the premises to be making the judgement about the set.

You could see this as leading to or including a hasty generalisation, as you’ve made an assumption about the set X belongs in based on one quality.

Every lawyer must pass the bar exam.
[x] has passed the bar exam.
Therefore [x] must be a lawyer.

The issue here is, they could be part of another set and not actively a lawyer. For example, they could teach law, be a judge or be a retired lawyer.

There could be other things that have excluded them from the set, perhaps they have been disbarred for malpractice, or perhaps they do not work in a law firm and are not practicing law.

• 8. Hasty Generalization Fallacy – YouTube

Is a False Dichotomy the Same as a Hasty Generalisation?

In a word, no. Sure, I could probably rephrase some false dichotomies in the form of a hasty generalisation and some hasty generalisations in the form of false dichotomies but that doesn’t mean they are the same thing.

If I can rephrase an instruction as a question, does that mean questions and instructions are the same thing or is language just flexible?

The same can be said for the set theory and conditional statements. All elephants have trunks could be rephrased as if it is an elephant then it has a trunk, but that isn’t the same thing.

Unfortunately, the conversation then took a bit of a weird turn.

Misunderstanding What a Conditional Statement Is

As often is the case with these sorts of online discussions, the explanations were “refuted” by more misunderstandings, instead of addressing the actual argument.

At this point, it felt like even the most basic ideas in logic were breaking down.

A conditional statement is one where there are requirements that need to be met for something to be true or the entailment to be the case. If you do your homework then I will let you play on the PlayStation. There is no guarantee you can play the PlayStation unless you meet the condition of doing homework.

A set statement is discussing the qualities (or at least one quality) of something that exists in a set. (With the exception of damage and defects) All elephants have 4 legs, trunks, can grow tusks and so on.

Being in the set guarantees those qualities. You can make a conditional statement like “if you’re in this set then you have the qualities of that set” or to use the elephant example, “If x is an elephant then x has a trunk” but that is not the same thing as saying all elephants have the aforementioned qualities.

The set statement is guaranteed, the conditional statement is… well.. conditional. If the condition is met then sure, the entitlement is guaranteed in a sound syllogism, but that is different to something that is describing the guaranteed qualities of something.

Syllogisms About Belief in God

Before a lot of the conversation happened that inspired the article, I tried giving an example that would be relatable to everyone in the thread. I figured it could be a nice way to tie off the explanation.

Whilst believing no gods exist entails not believing gods exist, the same isn’t necessarily true the other way around. You could be suspending judgement, unable to conclude either way e.g. not believing gods exist but also not believing gods don’t exist. This is what I imagine when someone says “I only lack belief!”

I think Christians would definitely disagree there, no? Aside from silly catchphrases like “You’re as atheist as I am in regard to other gods!” I think it fair to say that Christians believe in a God so are not non-theists or atheists.

In Conclusion

The fallacies of affirming the consequent and the undistributed middle can seem similar but apply to different things. It is easy to get them confused, especially in situations where you are only half paying attention.

A conditional statement does not have a guaranteed conclusion unless that condition is met.

Sets talk about guaranteed qualites of things within the set.

Both fallacies are errors in reasoning that could end up with the same conclusion but that doesn’t mean they are the same things.

Davidian

I’m Joe. I write under the name Davidian, not only because it is a Machine Head song I enjoy but because it was a game character I used to role-play that was always looking to better himself.

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