Context: Logical Induction is a framework that makes sense of intuitively plausible statements like “the probability that the 10101010th digit of π is odd is about 0.5”.
People often do this sort of informal reasoning about mathematical conjectures. Like “The Collatz conjecture has been checked up to 268, and held for all those—updating on this, I increase my likelyhood that the conjecture is true in general”.
Logical induction seems to provide, in principle, a set of rules that such updates should follow. How many of these rules are known?
Some example rules that seem very plausible (here all my variables are implicitly natural numbers):
The observation that ϕ(n0) is true does not decrease the likelyhood of ∀nϕ(n).
Updating on the observations ”ϕ(n) for all 0≤n≤N”, the probability of ∀nϕ(n) goes to 1 as N→∞
[Question] How much is known about the “inference rules” of logical induction?
Context: Logical Induction is a framework that makes sense of intuitively plausible statements like “the probability that the 10101010th digit of π is odd is about 0.5”.
People often do this sort of informal reasoning about mathematical conjectures. Like “The Collatz conjecture has been checked up to 268, and held for all those—updating on this, I increase my likelyhood that the conjecture is true in general”. Logical induction seems to provide, in principle, a set of rules that such updates should follow. How many of these rules are known?
Some example rules that seem very plausible (here all my variables are implicitly natural numbers):
The observation that ϕ(n0) is true does not decrease the likelyhood of ∀nϕ(n).
Updating on the observations ”ϕ(n) for all 0≤n≤N”, the probability of ∀nϕ(n) goes to 1 as N→∞
Do these hold for logical inductors?