Causation with Inductive/Deductive Logic

Question

I'm sofa-philosopher and I have a case I can't quite dismantle. My friend (let's call him John) just made a statement that I find philosophically weak, and I'd like to run it thru you guys.

So, John was infected with covid and had a mild course. This was our conversation:

John: "Luckily I was vaccinated, otherwise I might have died."

Me: "How do you know that your mild course was caused by the vaccination?"

John: "From statistics, of course!" (followed by a long list of statistics)

Me: "How sure are you?"

John: "100%!"

Me: "But you understand that statistics don't cause anything?"

John: "It's logical that vaccines helped me!"

Me: "It's a possibility, not a certainty. There can be many explanations; immune systems are complex and individual, most cases are mild so you were among the majority, and you don't have typical risk factors - you're not obese or diabetic for example - ..."

John: "Why don't you believe me?"

After this the exchange became unproductive.

My questions:

1. Who was using logic correctly here - me or John? (not looking for a medical debate, but the use of philosophy)
2. Which type of logic did John use? (I can't quite place his arguments as inductive nor deductive...)

That's it, thanks for your input.

/pom

Answer 1

This is mostly about probabilities and psychology.

Let's do a thought experiment. Suppose 100 helmet-wearing cyclists and 100 non-helmet-wearing cyclists crash. How many survive?

I made the numbers up, this isn't from any official statistics, but you get the idea. Now, one of the surviving cyclists wants to know what the chances are he was saved by his helmet.

This guy is not part of the 10 who died wearing the helmet, so that part of the sample is not considered. Among the 90 survivors who wore a helmet, 70 would have survived anyway without it. However, the difference is in the 20 (30-10) who would die if they didn't wear it, but stay alive if they did wear it.

Therefore, knowing that he was wearing a helmet and survived, the probability of his life being saved by the helmet is 20/(70+20) = about 22%.

The other question, "knowing that he survived, what is the probability he was wearing a helmet" would be 90/(70+90) = 56.2%, which is quite close enough to random in this case.

To illustrate the influence of the "treatment effect size", let's use a life jacket instead of a helmet. It is not very effective while crashing on a bike, so it only saved one guy who fell off a bridge into a lake.

In this case, on any random crash, the probability that the guy was saved by the life jacket is only 1.4%.

On the other side, the usual example is the parachute. There is always a story about one guy with freakish luck who survives jumping off a plane without a parachute, but that doesn't change the results very much.

Please ignore the fact you can actually know if the helmet saved your brain by examining it after the crash. It was just for the sake of using a non hysterically politicized example.

You can do the math with the actual covid data, though. You can get any result from "parachute" to "life jacket on a bicycle" depending on your friend's age and comorbidities, and also the variant of the day of course.

Your friend isn't using logic, he is simply exhibiting commitment-driven rationalization. So this isn't about philosophy and logic, rather about math (above) and psychology (below).

When humans commit to a costly action, they tend to rationalize after the fact that it was the right thing to do. This can take many forms, and it has pros and cons. It can soothe anxiety about having made the right choice, and make one happier about the choice made, however it makes one more prone to the sunk cost fallacy (among others).

Consider a cancer patient. The operation is successful and the tumor is removed, but there could be a tiny clump of cells remaining somewhere that wouldn't show up on a scan. If you want to know if it's there, you have to wait for it to grow enough so it shows up, which is not a good idea. So the patient is given chemo, loses his hair, and gets all sorts of nasty side effects. If the patient is cured, then it is impossible to know if there was actually some cancer left and the chemo destroyed it, or if there was none and he lost his hair for nothing. There is only probabilities, "we know that the patients who didn't get chemo died X% more". However, after spending six weeks throwing up, most people will forget all about the probabilities, and rationalize until they're absolutely convinced that it was 100% necessary. Can't blame them, and it makes the ordeal much more bearable. In fact, it would be quite rude to remind them that maybe they lost their hair for nothing, even if it is true.

The vax doesn't score very high on commitment (as long as you don't look too much into the side effects). However, it brings another type of commitment, of the tribal kind. Consider a hardcore football supporter: their team is the best, even when they lose, and you will never change their mind. Now, due to a tsunami of propaganda, we definitely have two teams/tribes: team Vax and team Unvax, who suitably blame each other for the mess we're in and fling poo at each other on twitter from the trenches they've dug. And they're quite firmly dug in and really stubborn. Therefore, I think an important and often neglected part of vax commitment is one of allegiance to a tribe.

This one is tough because it is very strong, and truth becomes completely irrelevant: it's pretty much "we eat the eggs from the big end, and the other guys we're at war with eat their eggs from the little end, those filthy heretics!" In these matters, whether the vax works or not is irrelevant. In fact, you will get more commitment (and therefore more rationalization) if it doesn't work at all, preferably with nasty side effects, because then you have to really believe it works in order to do it. Then it becomes like a ritual for the purpose of reinforcing belief. I would recommend reading about Festinger's experiment, "When prophecy fails".

So it is possible that your friend associates questioning the efficacy of the Holy Product with betrayal against their tribe, or even heresy. You're not going to logic your way out of this one.

Answer 2

To your first question:

There's a lot of heat surrounding covid and related issues, which makes it hard for people to reason effectively. When we abstract away from the specifics, the weakness of John's reasoning becomes much easier to see. Consider the two following statements:

• "I am 100% certain that I rolled a 4 on a six-sided die because the probability of rolling at least a 2 is 5/6"
• "I am 100% certain that I rolled a 1 on a six-sided die because the probability of rolling at least a 2 is 5/6"

It's true that there's a 5/6 chance of rolling at least 2, but does either of these statements sound at all compelling? There are several issues with this line of reasoning, one of which is the causality issue you've identified. The probability of any given outcome contributes zero causal force to the occurrence of that outcome. In reality, both the outcome and the probability are caused by the underlying mechanics of the system. This is vaguely similar to the Gambler's Fallacy.

A second big issue with this argument is that it seems to fundamentally misunderstand the numbers involved. Let's consider two hypothetical groups of people; group A and group B. There is a causal relationship between group membership and the probability of outcome X. For group A, the probability of outcome X is 1%. For group B, the probability of outcome X is 0.5%. If there are 1000 people in each group, 10 from group A will experience X and 5 from group B will experience X. 990 and 995 people from groups A and B respectively will not experience X. If we try to predict a person's group membership on the basis of not having experienced X, we end up with a roughly 49.9% chance of being in group A and a 50.1% chance of being in group B. In other words, the mere fact that the person did not experience X tells us virtually nothing whatsoever about why they didn't experience X, even though X is twice as likely for group A.

Another big issue here is the matter of 100% certainty. It is a mathematical fact that no amount of data can ever prove an empirical theory. This is called scientific underdetermination. From the Stanford Encyclopedia of Philosophy:

"If we consider any finite group of data points, an elementary proof reveals that there are an infinite number of distinct mathematical functions describing different curves that will pass through all of them. As we add further data to our initial set we will definitively eliminate functions describing curves which no longer capture all of the data points in the new, larger set, but no matter how much data we accumulate, the proof guarantees that there will always be an infinite number of functions remaining that define curves including all the data points in the new set and which would therefore seem to be equally well supported by the empirical evidence. No finite amount of data will ever be able to narrow the possibilities down to just a single function or indeed, any finite number of candidate functions, from which the distribution of data points we have might have been generated. Each new data point we gather eliminates an infinite number of curves that previously fit all the data... but also leaves an infinite number still in contention."

To your second question:

It may not make sense to try to categorize this type of reasoning because it's a broken argument. It could be presented as a deductive argument with at least one false premise. But I'd be inclined to call it an extremely weak inductive argument because of the focus on statistics. I'm not convinced that abductive reasoning is actually distinct from inductive reasoning.

Answer 3

Your friend is using logic correctly and you're not:

• Suppose the chance of dying after catching covid whilst unvaccinated: 1/50

• And the chance of dying after catching covid whilst vaccinated is: 1/10,000

Then he is correct to say:

"It's lucky, I was vaccinated otherwise I might have died".

Given the above, this would be turned into the statement:

"It's lucky, I was vaccinated as there is a higher probability of dying when unvacinnated than vacinnated".

And this is a statement deducible from the two statements above.