Logic 101 — Formal Fallacies

This week we’re diving into formal fallacies. Of all logic errors, these appear the most academic and technical. Don’t worry, all you need to understand comes from the following picture:

atheism diagram

A logical statement expresses if something, then something, or A ⇒ B (read if A, then B). For some reason, academic texts use p and q, so you’ll see things like p ⇒ q. We’ll stick with A and B, however.

Return to the picture and think about A and B. Clearly, if you are in A, you are in B. Thus, A ⇒ B, or A implies B, or if A, then B. All are equivalent statements, and you’ll see them all used. For our purposes, let’s stick with if A, then B or A ⇒ B.

See, logic isn’t really that hard, is it?

Technical Jargon

Let’s get some technical jargon out of the way. We can arrange A and B in other ways by negating one or the other (written with a tilde, like ~A), or switching the order to get:

  • If B, then A (B ⇒ A)
  • If NOT A, then NOT B (~A ⇒ ~B)
  • If NOT B, then NOT A (~B ⇒ ~A)

Those are known as the converse, inverse, and contrapositive of the original statement. Assuming A ⇒ B is true, are these other statements equally true?

As it turns out, only one of those is true. Which one? Look back at the picture.

  • If I am in B, then I am in A. Obviously not true. Anywhere in green I am in B, but not in A, so the statement is false.
  • If I am NOT in A, then I am NOT in B. Also not true for the same reason.
  • If I am NOT in B, then I am NOT in A. TRUE.

I think I can read your mind: you’re thinking “so what”? Let’s apply the previous and discover the first three logical problems we’ll cover.

Affirming the Consequent

A fancy name for using the using the converse of an argument and claiming it to be true:

  1. If A, then B
  2. B
  3. Therefore A

Why is it wrong? Because the statement A ⇒ B does not mean the only way to get to B. What if Z ⇒ B? Then B can be true without A being true, if Z is true.

The problem becomes much easier to spot with an example:

If it’s sunny tomorrow, I’m going to play golf (A ⇒ B, A is sunny day, B is golfing)

Affirming the consequent means if I play golf tomorrow, it’s sunny. That’s not necessarily true — it might be or might not be, we can’t tell from the original statement.

What if it rains, and I decide to play golf anyway?

The mistake of affirming the consequent comes from believing it’s the only way from A to B. And that’s not true.

Here’s another example you might have seen before:

The theory of evolution predicts fossils of a certain kind. We find these in the ground, therefore, evolution is true.

Same mistake. Let’s simplify it a bit: given Evolution ⇒ Fossils, proceeding to claim Fossils ⇒ Evolution.

That’s a logical error — it’s not the only way to get from A to B.

In the beginning God created the heaven and the earth (Genesis)

It also appears in global warming proponents and the (in)famous “hockey stick” graph. Global warming says rising Co2 means higher temperatures, we’ve seen those, thus the theory is true.

Affirming the consequent is sneaky, is it not?

Denying the Antecedent

Another fancy term, this time for claiming the inverse to be true (without proving it). Specifically: A ⇒ B also means NOT A ⇒ NOT B (or ~A ⇒ ~B) will be true as well.

  1. If A, then B
  2. Not A
  3. Therefore not B

Once again, looking at the picture at the beginning of the article shows this doesn’t work — it’s another error in critical thinking. Another example:

If I run a marathon, I will be thirsty. I did not run a marathon, therefore I am not thirsty.

That’s not true. I’m eventually going to be thirsty whether I do or do not run a marathon. I might get thirsty faster or need more water from running, but it’s not the only way to get from A to B.

It should be noted both the converse and inverse may be true. However, they can’t be assumed true from the original statement, they require proof of their own.

The reason the mistake can be hard to spot is because some of those arguments are true.

  1. Humans can speak (Humans ⇒ speak)
  2. It’s not human
  3. Therefore, it doesn’t speak

Denying the antecedent in this case turns out to be true, even though it’s a logical error. It works because (as far as we know) humans are the only creatures which speak — it really is the only way from A to BA is both a necessary (required) and sufficient condition for B.

Thus, A and B are equivalent — either both are true, or both are false.

Law of non-contradiction

Non contradiction simply means A can’t be NOT A, or “~A ⇒ A.”

That seems so obvious, surely nobody would claim such a thing? Knock Knock — go answer your door.

Standing before you appear two nicely dressed gentlemen, telling you they’re Christians too. Who is Jesus, you ask? Is He God?

Well, no, they reply, we’ve got this additional book telling us Jesus isn’t actually God.

So you don’t believe the Bible, you ask?

Oh yes, we do, they reply.

Gotcha! They violated the law of non-contradiction.

You see, it’s rather trivial to show Jesus is God from the Bible. See my article The Confusion of Religion for exactly how (I keep several copies by the door to use as a handout when they come by, and you’re invited to do so as well).

They stand before you claiming the Bible is true, while denying what it says. In short, “~A ⇒ A.”

As a side note (more in the previously mentioned article), do not engage in arguments about whether Jesus is—or is not—God. You simply want to point out they believe two things which can’t be true at the same time.

  • The Bible is true and authoritative.
  • Jesus isn’t God.

I never state Jesus is (or is not) God, only present the logical problem they find themselves in. For this purpose, we don’t really care if Jesus is God or not, only attempting to open their eyes to the horrible contradiction they’ve spun for themselves.

If I say the sky is orange, and you say it’s purple, at least one of us is wrong (maybe both). But we can’t both be right.

Non-Contradiction and Liberal/Progressive Theology

Liberal “Christians” fall in the same logical trap promoting the big-tent view (all world religions are equal) — even a casual reading of the world’s religions reveals many exclusive statements — they can’t all be right.

Liberal and progressive so-called Christians run into this problem many, many times. They all say the Bible is inspired by God — right up until they reject parts they don’t like.

Gotcha! The law of non-contradiction.

All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness (2 Timothy 3:16).

All means all, that’s all all means.

If liberal “Christians” believe the Bible, that means all of it is inspired, including the parts they want to throw out like the gospel and sin — errors originating from the cult of liberal theology.

Still not convinced liberals actually argue the absurd idea ~A ⇒ A? Consider the foundation of liberal and progressive theology — post-modern philosophy, from an article from “Christianity Today.”

The third kind of emerging postmodernity attracts all the attention. Some have chosen to minister as postmoderns. That is, they embrace the idea that we cannot know absolute truth, or, at least, that we cannot know truth absolutely. They speak of the end of metanarratives and the importance of social location in shaping one’s view of truth.

Minister as post-moderns? That means denying God’s truth. At its root, post-modernism argues absolute truth doesn’t exist, so you have your truth, and I have mine.

… a postmodernist is more comfortable with the both/and perspective, allowing multiple truths to exist in tension. It recognizes the significance of subjective reality on our understanding of truth, and as such, challenges more rigid doctrines, dogmas or policies that value uniformity of thought over pluralistic coexistence.

Buzzword Alert!

  • “subjective reality”
  • “multiple truths”
  • “pluralistic coexistence”

Notice the problem? They state absolute truth doesn’t exist, except for the one absolute truth is relative.


They’re arguing absolute truth does — and yet does not — exist, (or ~A ⇒ A) violating the law of non-contradiction.

Liberal theology makes no sense and is unbiblical for many reasons, but the primary reason not to bother with it is it fails the most basic of logical tests — the law of non contradiction.

Don’t waste any time with liberal, progressive, post-modern so-called Christianity. It’s not even worth five seconds of your time. It’s laughably absurd, denying God but lacking the guts to admit it … but that potluck sure was good, wasn’t it?


These mistakes occur frequently, and in fact most of us make these mistakes daily and the world doesn’t end. However, when trying to reason from one idea to another, it’s vital to use correct critical thinking and logic, or the conclusion fails to be valid.

That’s a long section. Additional articles in the series won’t be so technical; we’ll focus more on recognizing and correcting common logical errors.

Filed Under: Logic

Recommended Citation:
Yeager, Darrin "Logic 101 — Formal Fallacies" (2023-11-23 14:45),
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