Almost Surely Grice

Doctorow presented a post to this Club: "Probably Grice: Grice on probability". So here's a sequel.

From wiki, probability:

"In mathematics, a probability of an event A is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A).[6] An impossible event has a probability of 0, and a certain event has a probability of 1. However, the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely"".

If you then follow the link, you get to "almost surely" wiki -- abbreviation: "a.s.":

"In probability theory, one says that
an event happens almost surely (sometimes
abbreviated as a.s.) if it happens with
probability one. ... While there is no
difference between "almost surely" and "surely"
(that is, entirely certain to happen) in many
basic probability experiments, the distinction
is important in more complex cases
relating to some sort of infinity."

"Let"

Ω, F, P

"be a probability space."

"An event E in F happens almost surely
if P(E) = 1."

"Equivalently, we can say an event E happens almost surely if the probability of E not occurring is zero."

"An alternative definition from a
measure theoretic-perspective is that (since P
is a measure over Ω) E happens
almost surely if E = Ω almost everywhere."

---

"If an event is sure, then it
will always happen, and no
other event can possibly occur. If an
event is, however, almost sure, then
other events are theoretically possible in
a given sample space; however, as the cardinality
of the sample space increases, the probability
of any other event asymptotically converges toward
zero. Thus, one can never definitively say
for any sample space that other events will never occur, but can,
in the general case, assume this to be true."


It's still different with

analytic-apriori vacuous things like

"War is war".

-- (or less vacuously, Women are women).