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Basic Bayesian reasoning: a better way to think (Part 4)

December 15, 2014

Have you read the last several posts? In those posts we began the tale of Alice and Bob, a pair of murder suspects who recently started dating one another. Through their sordid tale, we'll examine Bayesian reasoning, the scientific method, and the so-called fallacy of "affirming the consequent".

Alice and Bob are going through a rough patch in their relationship. One day, Alice accuses Bob of infidelity, and they have this conversation:

Alice:
You spent the night at Carol's house last weekend! You're cheating on me with her! 

Bob:
What?! How do you figure that? I'm innocent! 

Alice:
If you're cheating on me with her, it makes perfect sense that you'd spend the night at her house! 

Bob:
Ha! You're "affirming the consequent". You've started from "if [cheating], then [night at Carol's house]", then concluded that "if [night at Carol's house], then [cheating]". This is a logical fallacy, and your argument is invalid. Cheating on you is not the only possible explanation for me spending the night at Carol's. There are other, perfectly innocent explanations - like the fact that Carol threw a party that ran late, and a bunch of us just crashed at her place for the night rather than risk driving home tired and drunk.

Now, let's pause the conversation here for the moment and assess the situation. So far, Bob's logic follows the example in the Wikipedia page on "affirming the consequent". And he certainly seems right - "affirming the consequent" is a fallacy in propositional logic, and Alice can't necessarily conclude that Bob cheated on her just because he spent a night at Carol's house. So, is Alice committing a logical fallacy? And therefore Bob is innocent? Let's continue and see:

Alice:
That party happened last weekend, when Carol knew I would be out of town! That makes perfect sense if Carol plotted to have you come without me! 

Bob:
That's ridiculous. You're still just "affirming the consequent". You've started from "if [Carol plotting], then [party on that weekend]", then concluded that "if [party on that weekend], then [Carol plotting]". I told you before, this is a logical fallacy. There are many other explanations for why the party might have happened on that weekend. 

Alice:
Whenever I ask Carol whether she's seeing anyone, she avoids the question! 

Bob:
That's more flawed reasoning. Do I have to explain it to you again? There are other possible explanations why Carol avoids that topic with you. That doesn't mean I'm cheating on you with her.

Alice:
But she was always very forthcoming about her dating life before. What would make her so reluctant to talk about it now? 

Bob:
I don't know. I suggest you ask her. Many innocent explanations are possible. You're still "affirming the consequent". You've started with a hypothesis - that I'm cheating on you with Carol - and then produced observations that fit with that hypothesis, then used those observations to justify the original hypothesis. That's like saying "The Bible is true because God wrote it, and it says that God exists. Therefore God exists". That kind of silly, circular reasoning is what happens when you "affirm the consequent", and you're using this logical fallacy over and over again to try to say that I've cheated on you.

Well, Bob certainly seems to be right. Alice can't, and shouldn't, conclude that Bob is cheating on her just because Carol is not talking about her dating life, or because the party happened on a certain weekend. After all, "affirming the consequent" is a logically fallacy, isn't it?

Alice:
Someone at the party saw you go into Carol's bedroom with her. 

Bob:
You're still "affirming the consequent", and it invalidates your conclusion. Your logic is flawed. There are perfectly innocent reasons to go into someone's bedroom. 

Alice:
With two bottles of wine. And you closed the door afterwards. 

Bob:
Still just "affirming the consequent". What, we're not allowed to drink wine at a party? We're not allowed to close the door when the music is loud out in the living room?

Alice:
This was at 10 pm, far before your normal bedtime, far before you're normally tired. 

Bob:
It was a crazy party. It wore me out fast. Why do you continue to "affirm the consequent"? Don't you see that you're still just starting with the idea that I'm cheating on you, then using that idea to interpret the events to justify itself? That's circular reasoning. You're saying that just because these are how things would play out IF I were cheating on you, therefore then I MUST be cheating on you. It's a logical fallacy, like I said many times, and you're just using it repeatedly.

Alice:
And you didn't come out from the bedroom until the next day. 

Bob:
Like I said, we got tired and decided to just sleep off the party rather than risk driving home drunk and exhausted.

Okay... hmm... I mean, "affirming the consequent" is still a logical fallacy, right? It's got a Wikipedia page and everything. How could Alice be right when she's committing this fallacy over and over? I mean... if you heard that your significant other went into a bedroom with someone else, along with two bottles of wine, and stayed behind closed doors until the next day, you wouldn't jump to any conclusions, right? Because you're a logical thinker and you don't want to commit a fallacy?

Alice:
My source from the party also tells me that it looked like you and Carol were making out before you went into her room.

Bob:
You know that eyewitnesses are unreliable. The living room was dark and your "source" was probably drunk as well. Or maybe your "source" is lying to break us up for his or her own ends. There's lots of possibilities, you can't conclude that I cheated on you from this, and you're only still "affirming the consequent" by bringing this up to say that I did.

Alice:
I also have this shopping receipt, dated the day before the party, for things that Carol bought. She purchased scented candles, those wine bottles we mentioned, and "sexy" lingerie. 

Bob:
What Carol does with her money and what lingerie she wears is none of my business. You're still using flawed logic, by starting from the idea that I'm cheating on you, to explain what Carol bought, then using that explanation to justify your initial assumption. 

Alice:
She also bought condoms. 

Bob:
I didn't know, I don't care, and it's not relevant. There are many reasons that Carol would buy condoms that have nothing to do with me cheating on you. 

Alice:
I found condoms of the same brand in Carol's trash dumpster after the party. They were used. 

Bob:
Are you crazy?! That's disgusting! Completely apart from your gross dumpster-diving, that doesn't prove anything. Those could have come from anywhere, thrown away by anyone. You're still trying to make everything fit into your preconceived notion that I've cheated on you. That's "affirming the consequent"! It's a logical fallacy! You're just repeating this fallacy over and over again!

Alice:
I had them tested at the lab. The DNA on the outside is a decently good match with samples from Carol's hair. 

Bob:
DNA matching is imperfect. There's thousands of people in this city that would also be a "good match" with that DNA sample. Even if were an "excellent" match there's still lots of people who would fit that criteria. Any one of them could have used the condom. And even if it WAS Carol, you can't conclude that I've cheated on you with her from just that. That would be "affirming the consequent"! 

Alice:
And the DNA on the inside is an excellent match with you. 

Bob:
LOGICAL FALLACY! Over and over again! Your reasoning is invalid! You're trying to go from "if A then B", to "if B then A"! That's circular logic! It's "affirming the consequent"! I did not cheat on you with Carol!

If you still believe Bob, then I have a bridge to sell you. The weight of evidence is overwhelming at this point. Bob did almost certainly cheat on Alice with Carol.

But what about "affirming the consequent"? Isn't Bob right that it's a logical fallacy? Isn't Alice's argument based entirely on using it over and over? What does Bayesian reasoning say about all this?

Now, Bayesian reasoning mirrors human common sense. It will never lead to a result that "normal" reasoning says is impossible. As I mentioned earlier, you don't actually need formal training to use it in your daily life, because its rules are just the rules of good thinking that's been refined to a mathematical precision. However, because of its precision and power over propositional logic, Bayesian reasoning can sometimes lead to surprising results for someone who's only versed in propositional logic. "Affirming the consequent" is one such result.

In Bayesian logic, "Affirming the consequent" is allowed in a mathematically precise way. You CAN relate "if A then B" to "if B then A". In Bayesian terms, where we assign probability values - P(A), P(B), P(A|B), et cetera - to all statements, "if A then B" can be expressed as P(B|A), and "if B then A" becomes P(A|B). And these two probabilities are directly related to one another, as it is plainly written out in Bayes' theorem:

P(A|B) = P(B|A)  * P(A)/P(B)

Essentially, the two factors grow together. As P(B|A) gets bigger, so does P(A|B). As B becomes better explained by A, A becomes more likely given B. The more strongly the consequences of a hypothesis are affirmed, the more likely the hypothesis is to be true. As more events around Carol's party are explained by Bob cheating on Alice, it becomes more certain that Bob cheated based on these events. So each event - each instance of "affirming the consequent" - actually strengthens the hypothesis that Bob cheated on Alice with Carol. Far from dooming Alice's hypothesis because of its status as a "logical fallacy", it actually serves as evidence for Alice's accusation.

That's right: "affirming the consequent" does not invalidate its conclusion, instead it actually serves as evidence FOR that conclusion.

It is the very fact that Alice used "affirming the consequent" OVER AND OVER that made her case so strong. It's crucial to note that if she had made only one such argument, even if that argument was the one from DNA on the condom, her case would have been weak and she would have been wrong to come to her conclusion. But with each instance of "affirming the consequent" - each time Alice successfully showed that the events around Carol's party fit with Bob cheating on her - her case grew stronger. Therefore, "affirming the consequent" is a "logical fallacy" only insofar as it's not being used enough.

So if you see someone say that Bill Gates must own Fort Knox because he's rich, you can legitimately say that this is flawed reasoning, and call him out on "affirming the consequent". In this case, you'd be using that term as a proper logical fallacy and saying that this person conclusion is invalid. But if this person repeated similar arguments over and over - if he showed that Bill Gates was part of a secret cabal that controlled the U.S. government, and that Gates had regularly been inside Fort Knox, and that there were mysterious changes to his net wealth that matched perfectly with mysterious changes in the amount of gold in Fort Knox, and that a highly ranked government official anonymously said "Bill Gates owns Fort Knox.", then we might be getting somewhere. Each of these things would by itself could be dismissed by citing "affirming the consequent", but together, each instance of "affirming the consequent" counts as evidence, and adds up to a strong case.

So "affirming the consequent" can both serve as evidence and be a mistake. But, in Bayesian terms, how can you tell when it's a mistake? What is the genuine blunder in logic when that happens? As a mistake, "affirming the consequent" is the act of coming to the conclusion without enough evidence. It's coming to the conclusion without affirming enough consequents. Or more properly, it's concluding that P(hypothesis|evidence) is high, when P(evidence|hypothesis) is not yet large enough to compensate for P(hypothesis)/P(evidence). The solution to this issue, in part, is not to stop "affirming the consequent", but to do it more - to look for more evidence.

The reason that propositional logic doesn't, and can't, follow this reasoning is because it cannot distinguish between probability values of 1% or 99%. In propositional logic, a statement can only be true, false, or undecided. But "affirming the consequent" works in Bayesian reasoning by moving the probability value: it perhaps starts at 1% (very unlikely to be true), but then slides to 20% (unlikely to be true), then to 70% (likely to be true), and to 99% (very likely to be true) as you affirm more consequents, over and over. Propositional logic sees this and says "all I recognize in all these cases are undecided statements", and since 99% is not 100%, it will not let you say that the conclusion is true. This is why "affirming the consequent" is always a logical fallacy in propositional logic. But this really says more about the limits of propositional logic rather than reflecting true rationality.

How do you know when you've affirmed enough consequents? How many times to you have to "affirm the consequent" to be sure of your conclusion? Due to the difficulties associated with using Bayes' theorem in a real-world context, it may be hard or impossible to get actual numbers. But you have to at least walk through the equations to vaguely answer the question.

In particular, when you work through the equation it turns out that the most effective kind of evidence is that which could be affirmed by your hypothesis, but not by a rival hypothesis. "Affirming the consequent" is better than not affirming. Circular reasoning is better than contradictory reasoning. This is the essence of the odds form of Bayes' theorem, which shows the importance of comparing the hypotheses against one another. It has many important applications:

One such application is the scientific method. Bayesian reasoning is the logical framework that underlies the scientific method. Science, in part, relies on "affirming the consequent". Experimental verification of theoretical predictions serves as evidence for that theory. On the flip side, theories are falsified based on experiments as well. Both sides of that statement are together expressed in the odds form of Bayes' theorem. Between two competing theories or hypothesis, "affirming the consequent" is better than not affirming, and circular reasoning is better than contradictory reasoning.

Bayesian reasoning is also at the heart of presuppositional apologetics, which starts with the idea that God of the Bible - who is the basis for all rational thought - exists. It then "affirms the consequent" by verifying that the world does indeed bear the image of its Creator. Rival non-Christian worldviews cannot make the same affirmation, and therefore must borrow from the Christian worldview even in attacking it, thereby contradicting themselves. Of course, its critics have said that this approach is invalid because it "affirms the consequent", but I hope you now know better.

This reasoning is also the logical foundation for my blog here. I start with this fundamental postulate: God as revealed in Jesus Christ. I then "prove" that God exists by demonstrating that this postulate generates the universe - that is, by affirming the consequent.

This Bayesian reasoning is also the logical framework for my series of posts on how science itself - its axioms and long-term traits and properties - serves as strong evidence for Christianity. Because hypothesis should be measured against its rivals, I said that science is evidence for Christianity and against atheism. Of course, its critics accused me of "affirming the consequent" over and over again. By now, you should recognize this is as the mark of a strong argument, one with a great deal of evidence behind it. After all, "affirming the consequent" is a hallmark of science itself.

In all these areas, beware those who only cry "fallacy!", who will not state or test their hypothesis against yours, who only want to tear down arguments instead of building them. They pretend that their ignorance is strength, because they think that knowing nothing means they never have to affirm any consequents. They do not realize that this is actually the mark of profound weakness, and such know-nothing hypothesis can only survive by parasitically attaching itself to more established theories. But you should actively seek to find, build, critique, and refine your hypothesis. Rejecting a hypothesis is never an end in itself, but a step towards a better hypothesis. Remember that the devil comes to steal and to kill and to destroy. But it is God who creates.

We can now conclude by answering the questions I raised at the end of my last post. Yes, Bayesian reasoning allows for "affirming the consequent", and this actually serves as evidence FOR your conclusion. There is still a sense where "affirming the consequent" is a fallacy, which happens when you give a hypothesis too much credit based on a single instance of "affirming the consequent". But this only means that you haven't affirmed enough consequents. To escape this fallacy, you need to affirm more consequents with your hypothesis, while comparing it with its rival hypothesis. "Affirming the consequent" is a fallacy in propositional logic, but that's more indicative of propositional logic's inflexible limits rather than a reflection of actual rationality. In fact, "affirming the consequent" forms half of Bayes' theorem in odds form, which is the logical basis for the scientific method, presuppositional apologetics, and this very blog and the theories I put forth in it.


You may next want to read:
What is "evidence"? What counts as evidence for a certain position?
Science as evidence for Christianity (Summary and Conclusion)
"Proving" God's existence
Another post, from the table of contents

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