# Inductive strength & tracking requirement

An inductive inference is usually considered strong if its conclusion is probably true, given true premises. In Tracking Track Records, Peter Lipton (giving credit to Nozick) suggests that a strong inductive inference is such that the premises track the conclusion, meaning something like this, "had the conclusion been false, the evidence would have been different." (p. 186)

What's the difference?

• Probabilistic conditional only imposes requirements on the possible worlds where the premise is true, adding the tracking condition also imposes requirements on the worlds where the conclusion is false. In general, high probability of a conditional does not say much about the probability of its contrapositive, see contrapositive of probability. The tracking condition attempts to remedy that. Feb 27, 2021 at 21:59

Say that H = hypothesis, O = observation. To compare the cases clearly, let's draw a Venn diagram of the possible outcomes:

``````╭H────╮
│A ╭──┼─O╮
│  │B │  │
╰──┼──╯C │
D ╰─────╯
``````

There's a box for H which intersects with the box for O. I've labeled the four areas: A = H \ O, C = O \ H, B = H ∩ O, D = (H ∪ O)ᶜ.

Notation: \ is set difference; H \ O means "the parts of H that are not also in O." ᶜ is set complement; (H ∪ O)ᶜ means "all the possible outcomes not in H and not in O."

If H is highly likely given O, this means that P(H|O) = P(B) / P(B ∪ C) is large (close to 1).

If Peter Lipton's condition, "had H been false, O would have been different," holds, this means that P(O|Hᶜ) is low (close to 0). So this is saying that P(C) / P(C ∪ D) is small (close to 0).

Now, to see the difference, consider if the outer region D is very high probability (a very large region in the Venn diagram) which also means that H and O are, a priori, both unlikely. In this case, Lipton's condition P(C) / P(C ∪ D) is always going to be small, because the size of region D dominates the size of region C.

But increasing the size of region D has no effect on the condition that "the conclusion is probably true, given the premises"; P(H|O) is unchanged.

So, Lipton's condition as you've stated it would call an inductive inference strong, whenever both hypothesis and the observation are a priori unlikely, regardless of whether the hypothesis is related to the observation. This is probably unintended behavior. I would suggest that Lipton's condition is mistaken, or perhaps you've misinterpreted it.