popper-logic-scientific-discovery

54 FINITE SEQUENCES. ORDINAL SELECTION AND
NEIGHBOURHOOD SELECTION
probability 147
Let us suppose that the elements of a finite reference-class α are numbered
(for instance that a number is written on each button in the box), and
that they are arranged in a sequence, in accordance with these ordinal
numbers. In such a sequence we can distinguish two kinds of selection
which have special importance, namely selection according to the
ordinal number of an element, or briefly, ordinal selection, and
selection according to its neighbourhood.
Ordinal selection consists in making a selection, from the sequence α, in
accordance with a property β which depends upon the ordinal number
of the element (whose selection is to be decided on). For example β
may be the property even, so that we select from α all those elements
whose ordinal number is even. The elements thus selected form a
selected sub-sequence. Should a property γ be independent of an ordinal
selection according to β, then we can also say that the ordinal selection is
independent with respect to γ; or we can say that the sequence α is,
with respect to γ, insensitive to a selection of β-elements.
Neighbourhood selection is made possible by the fact that, in ordering the
elements in a numbered sequence, certain neighbourhood relations are
created. This allows us, for example, to select all those members whose
immediate predecessor has the property γ; or, say, those whose first
and second predecessors, or whose second successor, have the property
γ; and so on.
Thus if we have a sequence of events—say tosses of a coin—we have
to distinguish two kinds of properties: its primary properties such as
‘heads’ or ‘tails’, which belong to each element independently of its
position in the sequence; and its secondary properties such as ‘even’ or
‘successor of tails’, etc., which an element acquires by virtue of its
position in the sequence.
A sequence with two primary properties has been called ‘alternative’.
As von Mises has shown, it is possible to develop (if we are
careful) the essentials of the theory of probability as a theory of alternatives,
without sacrificing generality. Denoting the two primary properties
of an alternative by the figures ‘1’ and ‘0’, every alternative can be
represented as a sequence of ones and zeros.