9.1: Hypothetical Reasoning
Suppose I’m going on a picnic and I’m only selecting items that fit a certain rule. You want to find out what rule I’m using, so you offer up some guesses at items I might want to bring:
A Banana
An Egg Salad Sandwich
A grape soda
Suppose now that I tell you that I’m okay with the first two, but I won’t bring the third. Your next step is interesting: you look at the first two, figure out what they have in common, and then you take a guess at the rule I’m using. In other words, you posit a hypothesis. You say something like
Do you only want to bring things that are yellow or tan?
Notice how at this point your hypothesis goes way beyond the evidence. Bananas and egg salad sandwiches have so much more in common than being yellow/tan objects. This is how hypothetical reasoning works: you look at the evidence, add a hypothesis that makes sense of that evidence (one among many hypotheses available), and then check to be sure that your hypothesis continues to make sense of new evidence as it is collected.
Suppose I now tell you that you haven’t guessed the right rule. So, you might throw out some more objects:
Grapes
A key lime pie
A jug of orange juice
I then tell you that the first two are okay, but again the last item is not going with me on this picnic.
It’s solid items! Solid items are okay, but liquid items are not.
Again, not quite. Try another set of items. You are still convinced that it has to do with the soda and the juice being liquid, so you try out an interesting tactic:
An ice cube
Some liquid water
Some water Vapor
The first and last items are okay, but not the middle one. Now you think you’ve got me. You guess that the rule is “anything but liquids,” but I refuse to tell you whether you got it right. You’re pretty confident at this point, but perhaps you’re not certain . In principle, there could always be more evidence that upsets your hypothesis. I might say that the ocean is okay but a fresh water lake isn’t, and that would be very confusing for you. You’ll never be quite certain that you’ve guessed my rule correctly because it’s always in principle possible that I have a super complex rule that is more complex than your hypothesis.
So in hypothetical reasoning what we’re doing is making a leap from the evidence we have available to the rule or principle or theory which explains that evidence. The hypothesis is the link between the two. We have some finite evidence available to us, and we hypothesize an explanation. The explanation we posit either is or is not the true explanation, and so we’re using the hypothesis as a bridge to get onto the true explanation of what is happening in the world.
The hypothetical method has four stages. Let’s illustrate each with an example. You are investigating a murder and have collected a lot of evidence but do not yet have a guess as to who the killer might be.
1. The occurrence of a problem
Someone has been murdered and we need to find out who the killer is so that we might bring them to justice.
2. Formulating a hypothesis
After collecting some evidence, you weigh the reasons in favor of thinking that each suspect is indeed the murderer, and you decide that the spouse is responsible.
3. Drawing implications from the hypothesis
If the spouse was the murderer, then a number of things follow. The spouse must have a weak alibi or their alibi must rest on some falsehood. There is likely to be some evidence on their property or among their belongings that links the spouse to the murder. The spouse likely had motive. etc., etc., etc.
We can go on for ages, but the basic point is that once we’ve got an idea of what the explanation for the murder is (in this case, the hypothesis is that the spouse murdered the victim), we can ask ourselves what the world would have to be like for that to have been true. Then we move onto the final step:
4. Test those implications.
We can search the murder scene, try to find a murder weapon, run DNA analysis on the organic matter left at the scene, question the spouse about their alibi and possible motives, check their bank accounts, talk to friends and neighbors, etc. Once we have a hypothesis, in other words, that hypothesis drives the search for new evidence—it tells us what might be relevant and what irrelevant and therefore what is worth our time and what is not.
The Logic of Hypothetical Reasoning
If the spouse did it, then they must have a weak alibi. Their alibi is only verifiable by one person: the victim. So they do have a weak alibi. Therefore...they did it? Not quite.
Just because they have a weak alibi doesn’t mean they did it. If that were true, anyone with a weak alibi would be guilty for everything bad that happened when they weren’t busy with a verifiable activity.
Similarly, if your car’s battery is dead, then it won’t start. This doesn’t mean that whenever your car doesn’t start, the battery is dead. That would be a wild and bananas claim to make (and obviously false), but the original conditional (the first sentence in this paragraph) isn’t wild and bananas. In fact, it’s a pretty normal claim to make and it seems obviously true.
Let’s talk briefly about the logic of hypothetical reasoning so we can discover an important truth.
If the spouse did it, then their alibi will be weak
Their alibi is weak
So, the spouse did it
This is bad reasoning. How do we know? Well, here’s the logical form:
If A, then B
B
Therefore, A
This argument structure—called “affirming the consequent”—is invalid because there are countless instances of this general structure that have true premises and a false conclusion. Consider the following examples:
If I cook, I eat well
I ate well tonight, so I cooked.
If Eric runs for student president, he’ll become more popular.
Eric did become more popular, so he must’ve run for student president.
Maybe I ate well because I’m at the finest restaurant in town. Maybe I ate well because my brother cooked for me. Any of these things is possible, which is the root problem with this argument structure. It infers that one of the many possible antecedents to the conditional is the true antecedent without giving any reason for choosing or preferring this antecedent.
More concretely, affirming the consequent is the structure of an argument that states that a) one thing will explain an event, and b) that the event in question in fact occurred, and then concludes that c) the one thing that would’ve explained the event is the correct explanation of the event.
More concretely still, here’s yet another example of affirming the consequent:
My being rich would explain my being popular
I am in fact popular,
Therefore I am in fact rich
I might be popular without having a penny to my name. People sometimes root for underdogs, or respond to the right kind of personality regardless of their socioeconomic standing, or respect a good sense of humor or athletic prowess.
If I were rich, though, that would be one potential explanation for my being popular. Rich people have nice clothes, cool cars, nice houses, and get to have the kinds of experiences that make someone a potentially popular person because everyone wants to hear the cool stories or be associated with the exciting life they lead. Perhaps, people often seem to think, they’ll get to participate in the next adventure if they cozy up to the rich people. Rich kids in high school can also throw the best parties (if we’re honest, and that’s a great source of popularity).
But If I’m not rich, that doesn’t mean I’m not popular. It only means that I’m not popular because I’m rich .
Okay, so we’ve established that hypothetical reasoning has the logical structure of affirming the consequent. We’ve further established that affirming the consequent is an invalid deductive argumentative structure. Where does this leave us? Is the hypothetical method bad reasoning ?!?!?!? Nope! Luckily not all reasoning is deductive reasoning.
Remember that we’re discussing inductive reasoning in this chapter. Inductive reasoning doesn’t obey the rules of deductive logic. So it’s no crime for a method of inductive reasoning to be deductively invalid. The crime against logic would be to claim that we have certain knowledge when we only use inductive reasoning to justify that knowledge. The upshot? Science doesn’t produce certain knowledge—it produces justified knowledge, knowledge to a more or less high degree of certitude, knowledge that we can rely on and build bridges on, knowledge that almost certainly won’t let us down (but it doesn’t produce certain knowledge).
We can, though, with deductive certainty, falsify a hypothesis. Consider the murder case: if the spouse did it, then they’d have a weak alibi. That is, if the spouse did it, then they wouldn’t have an airtight alibi because they’d have to be lying about where they were when the murder took place. If it turns out that the spouse does have an airtight alibi, then your hypothesis was wrong.
Let’s take a look at the logic of falsification:
If the spouse did it, then they won’t have an airtight alibi
They have an airtight alibi
So the spouse didn’t do it
Now it’s possible that the conditional premise (the first premise) isn’t true, but we’ll assume it’s true for the sake of the illustration. The hypothesis was that the spouse did it and so the spouse’s alibi must have some weakness.
It’s also possible that our detective work hasn’t been thorough enough and so the second premise is false. These are important possibilities to keep in mind. Either way, here’s the logical form (a bit cleaned up and simplified):
If A, then B
Not B
Therefore not A
This is what argument pattern? That’s right! You’re so smart! It’s modus tollens or “the method of denying”. It’s a type of argument where you deny the implications of something and thereby deny that very thing. It’s a deductively valid argument form (remember from our unit on natural deduction?), so we can falsify hypotheses with deductive certainty: if your hypothesis implies something with necessity, and that something doesn’t come to pass, then your hypothesis is wrong.
Your hypothesis is wrong. That is, your hypothesis as it stands was wrong. You might be like one of those rogue and dogged detectives in the television shows that never gives up on a hunch and ultimately discovers the truth through sheer stubbornness and determination. You might think that the spouse did it, even though they’ve got an airtight alibi. In that case, you’ll have to alter your hypothesis a bit.
The process of altering a hypothesis to react to potentially falsifying evidence typically involves adding extra hypotheses onto your original hypothesis such that the original hypothesis no longer has the troubling implications which turned out not to be true. These extra hypotheses are called ad hoc hypotheses.
As an example, Newton’s theory of gravity had one problem: it made a sort of wacky prediction. So the idea was that gravity was an instantaneous attractive force exerted by all massive bodies on all other bodies. That is, all bodies attract all other bodies regardless of distance or time. The result of this should be that all massive bodies should smack into each other over time (after all, they still have to travel towards one another). But we don’t witness this. We should see things crashing towards the center of gravity of the universe at incredible speeds, but that’s not what’s happening. So, by the logic of falsification, Newton’s theory is simply false.
But Newton had a trick up his sleeve: he claimed that God arranged things such that the heavenly bodies are so far apart from one another that they are prevented from crashing into one another. Problem solved! God put things in the right spatial orientation such that the theory of gravity is saved: they won’t crash into each other because they’re so far apart! Newton employed an ad hoc hypothesis to save his theory from falsification.
Abductive Reasoning
There’s one more thing to discuss while we’re still on the topic of hypothetical reasoning or reasoning using hypotheses. ‘Abduction’ is a fancy word for a process or method sometimes called “inference to the best explanation. The basic idea is that we have a bunch of evidence, we try to explain it, and we find that we could explain it in multiple ways. Then we find the “best” explanation or hypothesis and infer that this is the true explanation.
For example, say we’re playing a game that’s sort of like the picnic game from before. I give you a series of numbers, and then you give me more series of numbers so that I can confirm or deny that each meets the rule I have in mind. So I say:
20, 30, 40
And then you offer the following series (serieses?):
2, 3, 4
12, 22, 32
60, 90, 120
Each of these series tests a particular hypothesis. The first tests whether the important thing is that the numbers start with 2, 3, and 4. The second tests whether the rule is to add 10 each successive number in the series. The third tests a more complicated hypothesis: add half of the first number to itself to get the second number, then add one third of the second number to itself to get the third number.
Now let’s say I tell you that only the third series is acceptable. What now?
Well, our hypothesis was pretty complex, but it seems pretty good. I can infer that this is the correct rule. Alternatively, I might look at other hypotheses which fit the evidence equally well: 1x, 1.5x, 2x? or maybe it’s 2x, 3x, 4x? What about x, 1.5x, x\(^2\)? These all make sense of the data, but are they equal apart from that?
Let’s suppose we can’t easily get more data with which to test our various hypotheses. We’ve got 4 to choose from and nothing in the evidence suggests that one of the hypotheses is better than the others—they all fit the evidence perfectly. What do we do?
One thing we could do is choose which hypothesis is best for reasons other than fit with the evidence. Maybe we want a simpler hypothesis, or maybe we want a more elegant hypothesis, or one which suggests more routes for investigation. These are what we might call “theoretical virtues”—they’re the things we want to see in a theory. The process of abduction is the process of selecting the hypothesis that has the most to offer in terms of theoretical virtues: the simplest, most elegant, most fruitful, most general, and so on.
In science in particular, we value a few theoretical virtues over others: support by the empirical evidence available, replicability of the results in a controlled setting by other scientists, ideally mathematical precision or at least a lack of vagueness, and parsimony or simplicity in terms of the sorts of things the hypothesis requires us to believe in.
Confirmation Bias
This is a great opportunity to discuss confirmation bias, or the natural tendency we have to seek out evidence which supports our beliefs and to ignore evidence which gets in the way of our beliefs. We’ll discuss cognitive biases more in Chapter 10, but since we’re dealing with the relationship between evidence and belief, this seems like a good spot to pause and reflect on how our minds work.
The way our minds work naturally, it seems, is to settle on a belief and then work hard to maintain that belief whatever happens. We come to believe that global warming is anthropogenic—is caused by human activities—and then we’re happy to accept a wide variety of evidence for the claim. If the evidence supports our belief, in other words, we don’t take the time or energy to really investigate exactly how convincing that evidence is. If we already believe the conclusion of an inference, in other words, we are much less likely to test or analyze the inference.
Alternatively, when we see pieces of evidence or arguments that appear to point to the contrary, we are either more skeptical of that evidence or more critical of that argument. For instance, if someone notes that the Earth goes through normal cycles of warming and ice ages and warming again, we immediately will look for ways to explain how this warming period is different than others in the past. Or we might look at the period of the cycles to find out if this is happening at the “right” time in our geological history for it not to be caused by humankind. In other words, we’re more skeptical of arguments or evidence that would defeat or undermine our beliefs, but we’re less skeptical and critical of arguments and evidence that supports our beliefs.
Here are some questions to reflect on as you try to decide how guilty you are of confirmation bias in your own reasoning:
Questions for Reflection:
1. Which news sources do you trust? Why?
2. What’s your process for exploring a topic—say a political or scientific or news topic?
3. How do you decide what to believe about a new subject?
4. When other people express an opinion about someone you don’t know, do you withhold judgment? How well do you do so?
5. Are you harder on arguments and evidence that would shake up your beliefs?