Generics, Frequency Adverbs, and Probability

to infinite sequences, so long as these sequences are “sufficiently” long.
2.3 Branching Time
Intuitively, when we make a probability judgment, we consider not only the
sequence we have actually observed, but possible forms this sequence might
take in the future. It is, therefore, particularly attractive to investigate probability
judgments in a framework which regards time as nonlinear, or branching.
That is to say, for any given time there is more than one possible future.
There is a course of time where the world is destroyed in the year 2000, there
is a course of time where you become the President of Dalmatia, there is a
course of time where I never finish writing this paper, and so on.
I will refer to a linear course of time as a history. 7
History H continues
history H ′ , written as H ′ ❁ H, iff H ′ forms an initial segment of H.
Each sequence can be taken to represent one possible history. Hence I
propose the following informal definition of probability:
Definition 2 (Probability)
P (φ|ψ) = l iff for every admissible history H and every ɛ > 0, there is
a history H ′ ❁ H, s.t. for every history H ′′ , H ′ ❁ H ′′ ❁ H, the limiting
relative frequency of φs among ψ will be within ɛ of l.
For example, given a fair coin, for every history H and for every ɛ >
0 there is some initial segment of H where the relative frequency may be
7 The term is due to Thomason (1970).
20