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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 P - Proportion

 

Proportion: Ratio of the measure of a part to the measure of the whole it is part of; ratio of the cardinal number of a subset to the cardinal number of the set it is subset of.

The former definition may be regarded as the more general case, but the latter definition will be used in this lemma. The reason to include the lemma is that the theory of proportions is both elementary and a good basis for elementary probability theory.

The theory of cardinal numbers of sets can be seen as based on two assumptions:

  • A1. Two sets have the same cardinal number if and only if there is a 1-1 function between them.
  • A2. A set that is the union of a number of exclusive subsets has a cardinal number equal to the sum of the cardinal numbers of all these subsets.

In particular, using '#(X)=x' as short for 'the cardinal number of X is x' and using Xi and Xj as conventions to indicate that these are subsets of X, A2 can be used to infer the following properties of cardinal numbers of sets:

(1) #(Xi)+#(~Xi)=#(X)

for Xi and its complement ~Xi are exclusive subsets of X that when united form X, and thus (1) follows from A2.

(2) #(Xi) = #(XiOXj) + #(XiO~Xj)

This follows for a similar reason from A2: (XiOXj) and (XiO~Xj) are exclusive and when united are equal to (Xi).

Now we can introduce 'p(Xi)' for 'the proportion of Xi in X' and 'p(Xi|XJ)' for 'the proportion of Xi in Xj' and define (in fact set-theoretically speaking taking p(.) as a function from the subsets of X to the rational numbers, just as #(.) set-theoretically speaking is a function from the subsets of X to the natural numbers including 0):

(3) p(Xi) = #(Xi) : #(X)
(4) p(Xi|Xj) = #(XiOXj) : #(Xj)

Fundamental theorems

Here it makes sense to insert some statements of simple theorems with sketches of proofs that hold given the above stipulations concerning p and #. In fact all proofs are quite trivial. In what follows D indicates the whole domain of discourse, while Di and Dj indicate arbitrary subsets of D.

T1: #(~D) = 0
Pr: By (1) putting D for D
i (since D is its own subset).

T2: #() = 0
Pr: #() = #(D
iO~Di) by A1 since = DiO~Di and #DiODi=#Di by A1 since DiODi=Di.
Therefore as #Di = #(Di
ODi) + #(DiO~Di) by A2, it follows # = 0.

T3: p(Di|Dj) = p(Di
ODj) : p(Dj)
Pr: p(Di|Dj) = #(Di
ODj):#(Dj) by (4) which = (#(DiODj):#(D)):(#(Dj):#(D)) by algebra which = p(DiODj):p(Dj) by (4).

T4: p(Di) = p(Di|D)
Pr: (Di) = #Di : #D by (3) which since Di
OD=Di turns to #DiOD : #D = p(Di|D) by (4).

These theorems allow the surrection of something that is formally very much like elementary probability theory, in the guise of proportions.

That the theory of proportions is formally very much like elementary probability theory is so, but there are some niggles related to the interpretation of probability, that in the context of the present lemma must have the form of "probability is proportion plus a certain condition (or several) that enables inferences of probabilities from proportions".

In the context of Personal Probability, such a condition for a person a is the following, where S and S' are sets (respectively population and sample); K is the knowledge a possesses; and Si is a subset of S and S'j a subset of S':

prob(a,f,S,S') =d f: S 1-1 S' & ~(ETeK)(ESi)(ES'j)(pr(a,T) & T |- p(Si) rel p(S'j))

That is, a has a probability for S if and only if f is a partial 1-1 surjective function that maps a set S (population) to a set S' (sample) that is such that a has no minimally credible knowledge T that allows the valid inference that the proportion in the population differs from the proportion in the sample.

The reason that f is partial (not all members of S are or need to be mapped to the sample set) is that this is not needed, not necessary, and often not possible. (The last for example if the population is infinite, very large or extends to the future.) And it is surjective because every element in the sample should identify some member in the population.

The reason f is 1-1 is that the elements in the sample set (say a lottery with numbers that refer to elements in the population set) must uniquely identify the elements in the population.

And y e f(Si) IFF f-1(y) e Si - and note that f-1(.) is functional since f is 1-1.

As indicated, one way to obtain the above is by setting up a lottery in which the tickets identify elements of the population.

There is a more complete, if not fully completed, sketch of probability on the basis of proportion in Proportional Probability.

 


See also: Freudenthal, Probability, Proportional Probability


Literature:

Pollock, Spiegel

 Original: Auf 28, 2005                                                Last edited: 12 December 2011.   Top