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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 P - Proportional Probability

 

Proportional Probability: 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 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 elements s'.

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*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≥   &   T |- p(Si) rel p(S'j) )
  • Inductive Condition:
    T prel P --> ( P rel Q IFF Q rel P | T )

In words:

  • 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 empirical probabilities.

Thus, given a theory of proportions (or probability theory la Kolmogorov):

One has theoretical probabilities by adding AC (to known proportions).
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 monotonicity for (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, section 6)

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.)

And the 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 (consistently).
 


See also: Abduction, Inference, Personal Probability, Problem of Induction, Theory


Literature:

Maartensz

 Original: Dec 11, 2007                                                Last edited: 12 December 2011.   Top