Induction and a probability formula

I thought about how induction relates to the probability of something, and I came up with this formula. Induction is here taken as repeated confirming observations of some theory or hypothesis.

Formula. When the amount of confirmed observations, O, of a hypothesis goes towards infinity, then the probability of the hypothesis given the observations, Pr(H|O), goes towards 1 unless defeated by other evidence.

The last clause ‘unless defeated by other evidence’ is important, as without it the conditional would be false. It could be that some hypothesis correctly predicted an increasing amount of observations but that it also mistakenly predicted an increasing amount of other observations. These mistaken predictions would be very strong evidence against the hypothesis (falsification), and the probability of the hypothesis would not rise even though there was more and more evidence in form of confirmed observations for it.

It is said that a single false prediction of a hypothesis disproves it. (It is a counter-example to a universal (i.e. A-type) proposition.) That’s correct but if we have very strong evidence for some hypothesis and we think we have found some evidence against it (i.e. a false prediction) then it is very probably that the evidence against it is bogus in some way (i.e. is not evidence against it at all). This sounds like Hume’s maxim doesn’t it? It’s more probably that the observation (‘testimony’) against the hypothesis is bogus than the hypothesis being bogus.

Of course, if we were to find more and more evidence against the hypothesis, then at some point the probability that the hypothesis would be false will be greater than the probability that all the evidence is bogus (or misleading, as it is also called).

Mathematical representation

If anyone knows how to properly formalize this formula in mathematic symbols, then please inform me. I tried using a limit function but couldn’t figure it out. Basically I want to express the idea that when the amount of O goes toward infinity, then the probability of H given O goes toward 1. The ‘given O’ here is the set of all the observations. The clause about ‘unless defeated by other evidence’ can be expressed in propositional logic by including it into the antecedent proposition in the conditional; “If there is no other defeating evidence and the amount of confirmed observations goes toward…, then …”

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