Theory of probability based on the theory
of proportions plus additional assumptions.
For the moment this is a fast and brief sketch:
Suppose one has the theory of proportions (or PT à la Kolmogorov),
that is one has a formal framework like elementary probability theory.
Suppose one adds a person a and a's presumptive knowledge K(a) and
the following possible definitions and assumptions, perhaps using PP
from proportions, and assuming a population (domain) S with subsets Si
and elements s from which a may have samples S' with subsets S'i and
Also, I use an abbreviated notation for probabilities, with
"T=x" for "pr(T)=x" etc.: wherever there are propositional variables
between (in)equalities, one has a statement of probabilities involving
these variables and (in)equalities.
First some definitions that are needed:
T pcons S =d (K&T |- S) & ¬(K&¬T |- S)
T prel S =d S|T > S|¬T
T rel S =d ¬(T&S =
T irr S =d S|T = S|¬T
Q irr P | T =d P|Q&T = P|T
These allow the statement of three conditions:
- Abductive Condition:
T = x IFF Q = x & (S)(T pcons S & T prel S --> Q<=S)
- Random Condition:
¬(ETeK)(ESi)(ES'j)( T≥½ & T |- p(Si) rel p(S'j) )
- Inductive Condition:
T prel P --> ( P rel Q IFF Q rel P | T )
- AC: The probability of a
theory is the probability of its lowest relevant proper consequence.
- RC: There is no credible
knowledge that the proportions of sample and population are mutually
relevant or dependent.
- IC: Everything that is relevant
for a prediction of a theory iff it is also relevant for it on that theory.
Also, starting from beliefs about proportions (i.e. cardinalities of
sets) in the domain a.k.a. population:
- Theoretical probabilities
concern beliefs about proportions of populations based on theories.
- Empirical probabilities
concern beliefs about proportions of populations based on samples.
- Inductive probabilities
concern revised theoretical probabilities on evidence about relevant
Thus, given a theory of proportions (or probability theory à la
One has theoretical probabilities by adding AC (to known
One has empirical probabilities by adding RC (to known samples).
One has inductive probabilities by adding IC (to known experiments).
That is, one has minimal proper theoretical probabilities from AC;
one has fair samples from RC; and one has
(ir)relevancies from IC.
Speaking schematically and generally:
One starts with a domain of kinds of things about which one has
beliefs and theories, notably about cardinalities, and perhaps some
known proportions. One may use these with the AC to obtain probabilities
for theoretical beliefs, and one may use these with the RC to obtain
empirical probabilities for kinds one can sample. And if one obtains
experimental knowledge that some relevant condition has been verified or
falsified, one may use the IC to abstract from irrelevant circumstances
and obtain new theoretical probabilities from previous ones.
One needs the AC because one needs a way to attribute probabilities
to theories; one needs the RC because one needs to have proper samples
for evidence about the population; and one needs the IC to find new
theoretical probabilities, for one must be able to abstract from
irrelevant cirumstances (there always are).
It is noteworthy that the IC can be derived from the assumption that
theories must have true consequences only, but that it can be guaranteed
to be true only for specific known cases, for which reason it must
remain an assumption for any arbitratry condition. (See:
Problem of Induction,
Note all proportions and probabilities are personal (beliefs)
and may be socially shared, or not. Sharing implies agreements on at
least some empirical and theoretical procedures or assumptions, which
then may imply further agreements by logic or by agreed upon evidence.
Also, when all starts from a person's beliefs about the cardinalities
of sets, as it does with beliefs about proportions, it is all clearly
and simply relativized to both personal beliefs and to actual cardinalities
in the domain, that may be rationally established and investigated. (If
the last is not the case - "How many angels can dance on the tip of a
needle?"; "How many universes are there besides the one in which we
are?" - there are no rational probabilities at all.)
difference between the present approach to probability, which is
personal, and a subjective approach to probability, is that the present
approach concerns personal beliefs about actual proportions of some
kind, that may be false or true, whereas the subjective approach assigns
a probability to any statement a person is willing to bet on