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
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
- 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:
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)
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|Xi)' for 'the
proportion of Xi in Xj' and define
(3) p(Xi) = #(Xi) : #(X)
(4) p(Xi|Xj) = #(XiOXj) : #(Xj)
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 Di (since D is its own
T2: #(Ø) = 0
Pr: #(Ø) = #(DiO~Di)
by A1 since Ø = DiO~Di
by A1 since DiODi=Di.
Therefore as #Di = #(DiODi)
by A2, it follows #Ø = 0.
T3: p(Di|Dj) = p(DiODj)
Pr: p(Di|Dj) = #(DiODj):#(Dj)
by (4) which = (#(DiODj):#(D)):(#(Dj):#(D))
by algebra which
T4: p(Di) = p(Di|D)
Pr: (Di) = #Di : #D by (3) which since DiOD=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.
This is formally so, but there are some niggles related to the
probability, that in the context of the present lemma must have the
form of "probability is proportion plus a certain condition 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:
prob(a,f,S,S') =df f: S 1-1 S' &
~(ETeK)( pr(a,T)≥½ & T |- p(xeSi|f(x)ef(Si))≠p(xeSi))
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