In a previous post, I argued that falsifiability is not the be-all and end-all of Science. There are valid scientific beliefs that are not falsifiable.
However, there is something to the idea that beliefs should be falsifiable. One way to make this precise is to use Bayes' Theorem. This is a rule which says how to update your probabilities when you get some new evidence E. It says that your belief in some idea X should be proportional to your prior probability (how strongly you believed in before the evidence), times the likelihood of having measured the new evidence given X. (You also have to divide by the probability of having measured the new evidence, but this is the same no matter what X is, so it doesn't affect the ratio of odds between two competing hypotheses X and Y. It's just needed to get the probabilities to add up to 1). As an equation:
We won't actually plug any numbers into this equation in this post. Instead, I'll just point out a general property which this equation has. Suppose you are about to perform an experiment. On average, you expect that your probability is going to be the same after the experiment as before.
For example, suppose you believe there is a 1/50 chance that there exists a hypothetical Bozo particle (I just made that up right now). And suppose you perform an experiment which has a 50% chance of detecting the Bozo if it exists. Just for simplicity in this example let's suppose there are no false positives: if you happen to see the Bozo, it leaves a trail in your particle detector which can't be faked.
There are two possible outcomes: you see the Bozo or you don't. In order to see the Bozo, it needs to (a) exist and (b) deign to appear, so you have a 1% chance of seeing it. In that case, the probability that the Bozo increases to 1.
On the other hand, you have a .99 chance of not seeing the Bozo. In that case, your probabilty ratio goes from 49:1 to 98:1 since the Bozo exists possibilities just got halved. This corresponds to a 1/99 probability that the Bozo exists.
On average, your final probability is . Miraculously, this is exactly the same as the intitial probability 1/50 of the Bozo existing! Or maybe it isn't so much of a miracle after all. On reflection, it's pretty obvious that this had to happen. If you could somehow know in advance that performing an experiment would tend to increase (or decrease) your belief in the Bozo, that would mean you that just knowing that the experiment has been done (without looking at the result) should increase or decrease your probability. That would be weird. So really, it had to be the same.
We call this property of probabilities Reflection, because it says that if you imagine yourself reflecting on a future experiment and thinking about the possible outcomes, your probabilities shouldn't change as a result.
Now Reflection has an interesting consquence. Since on average your probabilities remain the same, if an experiment has some chance of increasing your confidence in some hypothesis X, it must necessarily also have some chance of decreasing your confidence in X. And vice versa. They have to be in perfect balance.
This means, you can show that it is impossible for an observation to confirm a hypothesis, unless it also had some chance of disconfirming it. VERY ROUGHLY SPEAKING, we could translate this as saying that you can't consider a theory to be confirmed unless it could have been falsified by the data (but wasn't).
Even so, there are a number of important caveats. In some situations in which we can and should believe things which are, in various senses, unfalsifiable. This occurs either because (a) The Reflection principle doesn't rule them out, or (b) the Reflection principle has an exception and doesn't apply. Here are all the important caveats I can think of:
- It could be that the probability of a proposition X is already high (or even certain) before doing any experiments at all. In other words, we know some things to be true a priori. For example, logical or mathematical results (such as 2+2 = 4) can be proven with certainty without using experiments. Similarly, some philosophical beliefs (e.g. our belief that regularities in Nature suggest a similar underlying cause) are probably things that we need to believe a priori before doing any experiments at all.
Propositions like these need not be falsifiable. This does not conflict with Reflection, because that only applies when you need to increase the probability that something is true using new evidence. But these propositions start out with high probability.
- It could be that a proposition has no reasonable chance of being falsified by any future experiment, because all the relevant data has already been collected, and it is unlikely that we will get much more relevant data. Some historical propositions might fall into this category, since History involves unrepeatable events. Such propositions would be prospectively unfalsifiable, but it would still be true that they could have been falsified. This is sufficient for them to have been confirmed with high probability.
- Suppose that we call a proposition verified if its probability is raised to nearly 1, and falsified if its probability is lowered to nearly 0. Then it can sometimes happen that a hypothesis can be verifiable but not falsifiable. The Bozo experiment above is actually an example of this. There is no outcome of the experiment which totally rules out the Bozo, but there is an outcome which verifies it with certainty (*).
This doesn't contradict Reflection. The reason is that Reflection tells us that you can't verify a hypothesis without some chance of lowering its probability. But it doesn't say that the probability has to be lowered all the way to 0. In the Bozo case, we balanced a small chance of a large probability increase against a large chance of a smaller probability decrease.
The Ring Hypothesis was another example of this effect. We have verified the existence of a planet with a ring. Had we looked at our solar system and not seen a planet with a ring, this would indeed have made the Ring Hypothesis less likely. But not necessarily very much less likely. Certainly not enough to consider the Ring Hypothesis falsified.
- Suppose that, if X were false, you wouldn't exist. Then merely by knowing that you exist, you know that X is true. But X is unfalsifiable, because if it were false you wouldn't be around to know it.
For example, no living creature could ever falsify the hypothesis that the universe permits life. Even though it didn't have to be true. Nor could you (in this life) ever know that you just lost a game of Russian Roulette.
This type of situation is an exception to the Reflection principle. The arguments for Reflection assume that you exist both before and after the experiment. (You can also construct counterexamples to Reflection involving amnesia, or other such funny business.)
To conclude, these are four types of reasonable beliefs which cannot be falsified. It is a separate question to what extent these types of exceptions tend to come up in "Science" as an academic enterprise (as opposed to other fields). But I don't see any good reason why these exceptions can't pop up in Science.
(*) Footnote: Some fictitious person (let's call her Georgina) might say that the Bozo is still falsifiable since nothing stops us from doing the experiment over and over again, until the Bozo is either detected or made extremely improbable. Hence, Georgina would argue, the Bozo IS falsifiable.
My answer to Georgina is that it actually depends on the situation. Maybe the Bozo experiment can only be done once. Maybe (since I'm making this story up, I can say whatever I want) the Bozo can only be detected coming from a particular type of Supernova, and it will be millions of years before the next one. More realistically, maybe the Bozo is detected using its imprint on the Cosmic Microwave Background, and the phenomenon of Cosmic Variance means that you can't repeat the experiment (since there is only one observable universe, and you can't ask for a new universe). More realistically still, maybe the experiment costs 100 billion dollars and Congress can't be persuaded to fund it more than once.
Georgina might not like the last example very much, since she might say that all she cares about is that the Bozo is in principle falsifiable. Perhaps as a holdover from logical positivism, the Georginas of this world often talk as though this makes some kind of profound metaphysical difference. But it's not clear to me why we should care about falsifiability in principle. The only thing that really helps us is falsifiability in fact.
If a critical experiment testing the Bozo will not be performed until next year, for purposes of deciding what to believe now, we should behave in exactly the same way as if the experiment could never be done. Experiments can't matter until we do them.