Interpretations of Probability
First published Mon Oct 21, 2002; substantive revision
Sat Jul 7, 2007
‘Interpreting probability’ is a commonly used but
misleading name for a worthy enterprise. The so-called
‘interpretations of probability’ would be better called
‘analyses of various concepts of probability’, and
‘interpreting probability’ is the task of providing
such analyses.
Or perhaps better still, if our goal is
to transform inexact concepts of probability familiar
to ordinary folk into exact ones suitable for
philosophical and scientific theorizing, then the task
may be one of ‘explication’ in the sense of Carnap
(1950). Normally, we speak of interpreting a formal
system, that is, attaching familiar meanings to the
primitive terms in its axioms and theorems, usually
with an eye to turning them into true statements about
some subject of interest. However, there is no single
formal system that is ‘probability’, but rather a host
of such systems. To be sure,
Kolmogorov's
axiomatization
, which we will present shortly, has
achieved the status of orthodoxy, and it is typically
what philosophers have in mind when they think of
‘probability theory’. Nevertheless, several of the
leading ‘interpretations of probability’ fail to
satisfy all of Kolmogorov's axioms, yet they have not
lost their title for that. Moreover, various other
quantities that have nothing to do with probability do
satisfy Kolmogorov's axioms
, and thus are
interpretations of it in a strict sense: normalized
mass, length, area, volume, and indeed anything that
falls under the scope of measure theory, the abstract
mathematical theory that generalizes such quantities.
Nobody seriously considers these to be ‘interpretations
of probability’,
however, because they do not play the
right role in our conceptual apparatus. Instead, we
will be concerned here with various probability-like
concepts that purportedly do. Be all that as it may, we
will follow common usage and drop the cringing scare
quotes in our survey of what philosophers have taken to
be the chief interpretations of probability.