When we pose a question, it is usual that we want an answer. Sometimes the answer is clear, because it is defined to be what it is, or else is true as a matter of logic.

For example, what is 2 plus 2? If your number system defines the answer to be 4, then it is 4.

If I roll two standard dice, the highest total I can achieve is 12. So if I am asked what is the probability that I will roll a total of 14, the answer is zero.

For most questions, however, which are not true by definition or logic, there is no sure answer to any question, only various levels of probability.

Similarly, for any set of observations, the rule or set of rules that gave rise to these observations might not be clear. There may be a large number of different explanations which are consistent with the data.

For example, what rule gives rise to the number sequence 1,3,5,7? If we know this, it will help us to predict what the next number in the sequence is likely to be, if there is one.

Two hypotheses spring instantly to mind. It could be: 2n-1, where n is the step in the sequence. So the third step, for example, gives 2×3-1 = 5. If this is the correct rule generating the observations, the next step in the sequence will be 9 (5×2-1).

But it’s possible that the rule generating the number sequence is: 2n-1 + (n-1)(n-2)(n-3)(n-4). So the third step, for example, gives 2×3-1 + (3-1)(3-2)(3-3)(3-4) = 7. In this case, however, the next step in the sequence will be 33.

So if this is all the information we have, we have two different hypotheses about the rule generating the data. How do we decide which is more likely to be true? In general, when we have more than one hypothesis, each of which could be true, how can we decide which one actually is true?

For the answer, we need to turn to some basic principles of scientific enquiry. I list these as Epicurus’ Principle, Occam’s Razor, Bayes’ Theorem, Popperian ‘Falsifiability’; Solomonoff Induction and the Vaughan Williams ‘Possibility Theorem.’

To address these, and how they contribute to a grand unified theory of scientific enquiry, is beyond the scope of this post, but I can at least provide a basic explanation of the terms.

Epicurus’ Principle is the idea that if there are a number of different possible truths, we should keep open the possibility that any of them might be true until we are forced by the evidence to do otherwise. Otherwise stated, it is the maxim that if more than one theory is consistent with the known observations, keep them all.

Occam’s Razor is the idea that the theory which explains all (or the most) and assumes the least is most likely. This is totally consistent with Epicurus’ Principle, with the additional insight that a simpler theory consistent with known observations is more likely to be true.

Bayes’ Theorem is the idea that the likelihood of a hypothesis being true is a combination of the likelihood of it being true before some new evidence arises and the likelihood of the new evidence arising if the hypothesis is true and if the hypothesis is false. Its most critical insight is that the probability of a hypothesis being true given the evidence is a very different thing to the probability of the evidence arising given that the hypothesis is true.

Popperian ‘Falsifiability’ is the idea that a scientific hypothesis should be testable and falsifiable. Otherwise stated, it notes that a single observational event may prove hypotheses wrong, but no finite sequence of events can verify them correct.

Solomonoff induction is the idea that the information contained in the various explanations consistent with known observations can in principle be reduced to binary sequences, and that the shorter the binary sequence the more likely that explanation of the observations is to be true.

The Vaughan Williams ‘Possibility Theorem’ states that: “If something that might or might not exist is unobservable, or is less likely to be observed, it is more likely to exist than if it can be observed (or is more likely to be observed) but is not observed.” This is critical when assessing how the probability of a hypothesis being true might be affected by information which potentially exists and is relevant but is missing because it is for whatever reason unobserved or unobservable.

Combining these principles into a unified framework can help identify the truth based on known and potentially missing observations.

That is the next step.