Interpretations of
Probability: There have been
many interpretations of what probability
is. I will sketch five, namely two more or less oldfashioned ones; two
currently fashionable ones; and my own theory of cardinal probability,
that is compatible with the last two.
Again, to treat the subject of the
interpretations of probability well, one
needs at least a book, and so I will limit myself to the four
interpretations that make some sense, and refer the reader to
Weatherford for a booklength survey of the field, followed by my own
theory, which will be considered in some more detail later on in this
paper.
Logical interpretation: The logical interpretation seems to be
the oldest interpretation if what probability is, and is often rendered
as "probability is the ratio of selected cases to possible cases". Thus,
the probability of throwing a 4 with an ordinary die is 1/6, and the
probability of throwing an even number with an ordinary die is 3/6.
As will be seen, there is
an underlying assumption that all possibilities that are distinguished
count for the same and as one, which much simplifies the treatment of
problems involving probability, but cannot easily or at all deal with
weighted dice or different and unpredictable lengths of life.
Empirical interpretation: The empirical interpretation soon
followed the logical interpretation, and tends to look at the actual
frequencies with which distinguished possibilities in fact happen as
information for what the probability of an event is.
This works well in
practice with subjects where one can easily establish frequencies or
samples, but this also makes it difficult
to say what probability really is, both for such things as have
frequencies, since these may change and anyway are partial information,
and for such things as have no frequencies, like unique events and
future events.
Both the logical and the
empirical interpretation are somewhat oldfashioned, though one still
meets the empirical interpretation in social statistics.
Objective interpretation: This can be best rendered in the form
of two claims, namely (1) there is real chance
in the world, in the form of chance processes and chance events in
physics, and real contingency in life and free choice and (2)
probability theory provides the tools to represent its basic properties.
This can be seen as
motivated by physics: According to quantummechanics there are real
chance processes in nature. Until the rise of quantummechanics, all
physical theories were deterministic, and probabilities only entered
because one nearly always has incomplete knowledge and samples of
populations. With the rise of quantummechanics this supposed
determinism of nature had to be given up.
Subjective interpretation: There are various subjective or
personal theories of probability. One way of rendering their intuitive
basis is in terms of two claims:
(1) Persons have their own personal estimates of probabilities, which,
if they are consistent, indeed behave according to probabilitytheory,
and (2) these probabilities can be used for
Bayesian confirmation.
This does justice to the
fact that different persons may have different estimates of what is the
probability of something, and enables each person to recalculate his
original probabilities when given new evidence.
The first claim can be
spelled out in quite a few different ways, based on different
considerations, but these will not occupy us here since we assume its
conclusion anyway.
It is especially the
second claim which makes subjective interpretations useful. The reason
is that while Bayes' Theorem
is a rather elementary theorem of formal
probability theory, applying Bayes' Theorem requires that one
has p(T),
i.e. the probability of a theory of which one desires to
recalculate the
probability of given new evidence about the predictions of the
theory, and this one does not have on the standard nonsubjective
interpretations, for whatever theories represent, these things
cannot be
counted like cherries, and anyway will at least start to be
largely
unknown for new theories.
The reason one does not
have this on the standard nonsubjective interpretations is a
fundamental lack of knowledge about the hypothesis T. But since one
does have this on subjective interpretations, one may make
any guess about the probability of any statement, provided only it is
consistent with one's further assumptions. The setback of this is that
if this is wholly subjective, one can in principle fix it so that almost
any evidence will have hardly any effect on it. Thus, not only need
subjective probabilities not be based on the evidence, but they also can
be chosen so extreme as to make almost any evidence have almost no
effect.
Cardinal interpretation:
The interpretation of probability I propose I call the cardinal
interpretation, because it rests on the existence of cardinal
numbers, which are guaranteed by nonprobabilistic assumptions, namely,
those given for extension and number and which exist anyway.
Hence there always will be
some probability for any statement, and this probability will exist
objectively because it derives from the cardinal numbers of the sets
that are involved.
One setback is that
normally one does not know the cardinal probability, though one can
normally establish evidence for such statements as represent things that
can be counted empirically, rather as in the empirical interpretation of
probability.
Another setback is that
one cannot count the things that are represented by a theory. The way to
solve that problem is to make an assumption about the probability of a
theory that is consistent with the rest of probability theory, and does
not depend on personal whim but on logic. It is what I called the
abductive condition: The probability of a theory T on background
knowledge K is the probability of its least probable known proper
consequence on K.
This amounts to a strengthening of the following theorem of any standard
formal probability theory:
(T)(Q) (T = Q
> p(T)<=p(Q)
)
That is: For any
statements T and Q, if T does entail (explain) Q, then the probability
of T is not larger than the probability of Q.
The abductive condition
strengthens this inequality to an equality in the case that Q is the
least probable of T's known consequences, and it does so to obtain a
probability for T  that then can be changed by any incoming evidence by
using Bayes' Theorem.
The cardinal
interpretation of probability is compatible with the objective
interpretation, and is like the subjective interpretation in enabling
the use of Bayesian confirmation, but it does not make this subjective,
though it does make this dependent on such evidence as one has,
including such consequences of the theory one has established.
Note it also has the
interesting consequence that wherever we have a domain of sets we have
implied probabilities for the sets, which exist as much as do the
cardinal numbers of these sets  but that very often we don't have
enough information to determine these cardinal numbers, and accordingly
the best we can do is to make a guess about it, and try to confirm or
infirm that guess by evidence.
