Conflict between epistemic and objective notions of probability §
In the former, to express a claim about a coin toss having a 50% chance of
being heads is to make a claim about a rational being’s own knowledge,
among other things.
Intuitively, we’d prefer to just say probability is a fact in the world,
about a given chance mechanism.
However, we often harbor metaphysical notions of determinism: i.e. with full
information before the toss (placement of coin in the hand, facts about the
coin-tossers brain, etc.), we could deduce the outcome of the toss.
This makes us believe that, objectively, the coin toss result being heads
is either 0 or 1. In general, there would be no non-extreme probabilities,
which is at odds with what makes probability theory useful.
Therefore, the desire to not step on determinism’s toes historically has led
to a dominance of epistemic notions of probability.
it would be remarkable if the ordinary claim that a coin toss is fair were
covertly a commentary on one’s own ignorance.
analogy: whether or not we are justified in believing something depends on
our epistemic relation to the world. However, this doesn’t mean that the
content of all our beliefs makes reference to our epistemic state.
But still, we then have to explain how to reconcile objective probability
with determinism.
Conflict between determinism and objective probability is an illusion due to a misconception about the content of judgments of probability. The misconception: §
Judgments of deduction viewed as just an extreme judgment of probability
(where deduced judgments have probability 1).
All beliefs have an associated credence and full belief is merely a special
case when that value is 1.
Example: “all balls in the urn are black, draw a ball, that ball will be
black” is just a special case with p=1 in “fraction p of the balls in the
urn are black, draw a ball, it has probability p of being black”
The 10-coin toss example below will show why the former is not special
case of the latter, as the latter has a certain ambiguity.
Unambiguous statement “10 coins are drawn, 5 of which are heads” §
If asked to compute the probability of it, we can do a calculation, but
we’re implicitly assuming some extra structure because the question is
ambiguous.
You get a different answer if we are informed the 10 coin tosses arose in
the context of a different experiment: “toss the coin repeatedly until you
get 5 heads in a row”.
Before we assumed the experiment was that the coin would be tossed 10 times
and then experiment would stop. This is just a different modal assumption,
but both interpretations are consistent with the factual statement in the
problem.
We can’t asses probability of a proposition until we embed it in a modal
structure.
The content of probability judgments are not propositional contents, but
rather propositions embedded in some procedural context.
They are not the types of things we can arrive at by deduction.
Thus, probabilitic reasoning and deduction are distinct modes of inference. §
Observing the coin is heads after the fact is no argument against the
purportedly probabilistic nature of the coin toss.
The reason why it’s not a good idea to bet against Laplace’s Demon is not
because the world has only objectively only extremal probability, but
because the demon is not using probabilistic reasoning at all (he might as
well be looking at the coin after the fact - he doesn’t have to assess
probabilities at all).
Like playing poker against an opponent with x-ray vision should not make
us believe objective probability does not exist.
Theoretical study of objective probabilities is back on the table.
relationship between inductive and deductive logic
E.g. Carnap’s inductive logics have a language with play a role of setting
up a procedural context in which it makes sense to ask for probabilities.
Language does not take such a role in deductive contexts.
E.g. a first order logic / deductive languages. If we make elementary
statements more specific, the deductive relations will not change, but
in an inductive language the probabilities may change dramatically.
What sentences to pick as elementary or basic is more important in an
inductive language