Propensity
- From: "Lance" <LanceGary@xxxxxxxxxxxxx>
- Date: 12 Jul 2006 06:26:08 -0700
Propensity
An academic from an out-of-the-way country like South Africa or New
Zealand always worries when he has to pronounce a name he has only seen
in print. Take the name of the American pragmatic philosopher C S
Peirce - I am told it is pronounced "purse", but who would have
guessed just from the spelling? Anyway perhaps the first propensity
account of probability is to be found in C S Peirce's writings
("Notes on the doctrine of chances"):
"I am, then, to define the meaning of the statement that the
_probability_, that if a die be thrown from a dice box it will turn up
a number divisible by three, is one-third. The statement means that the
die has a certain 'would-be'; and to say that the die has a
'would-be' is to say that it has a property, quite analogous to any
_habit_ that a man might have. Only the 'would-be' of the die is
presumably as much simpler and more definite than the man's habit as
the die's homogenous composition and cubical shape is simpler than
the nature of the man's nervous system and soul; and just as it would
be necessary, in order to define a man's habit, to describe how it
would lead him to behave and upon what sort of occasion - albeit this
statement would by no means imply that the habit _consists_ in that
action - so to define the die's 'would-be' it is necessary to
say how it would lead the die to behave on an occasion that would bring
out the full consequence of the 'would-be'; and this statement will
not of itself imply that the 'would-be' of the die _consists_ in
such behavior." (Peirce, 1910, pp.79-80).
Peirce goes on to describe "an occasion that would bring out the full
consequence of the 'would-be' - an infinite sequence of throws of
the die and the relevant behaviour of the die is that the appropriate
relative frequencies fluctuate around 1/3, gradually becoming closer to
this value.
Popper in his writings at different times offered three theories of
probability. His first account was a frequentist theory, but the later
two accounts are propensity theories. In his first (1957) propensity
theory of probability he disagrees with Peirce in analyzing the
'would-be' as a property of the die. Instead he analyses it as a
property of the die _and_ the conditions under which the die is thrown.
Popper offers some examples to support the idea that propensities are
properties of the set of conditions in which an event happens. Suppose
a coin is biased in favour of 'heads'. If we tossed it in the lower
gravitational field of the Moon the bias of the coin would have less
effect and the probability of obtaining heads when tossing the coin
would take on a lower value. This suggests an analogy between
probability and weight. We ordinarily think of weight as a property of
a body but it is really a relational property of a body with respect to
a gravitational field. Popper's second example is tossing a fair coin
onto a table in which a number of slots have been cut so we no longer
just have the two possibilities of 'heads' and 'tails', but
also a third, 'edge', in which the coin sticks in one of the slots.
Because 'edge' will have a finite probability the probabilities of
obtaining 'heads' and 'tails' must be reduced. The point of
this example is that not only do the probabilities of outcomes change
when the conditions are changed, but that even the nature of the
outcomes can change.
Both Popper and Peirce agree, however, in assigning the die a
disposition - a 'would-be' - and then distinguishing between
that disposition and the occasion that would bring out the full
consequence of the 'would-be'. This distinction allows Popper to
talk of probabilities as dispositions even on occasions where the full
consequences of the disposition are not manifested.
Popper of course was reacting against Von Mises' account of
probability. Von Mises had introduced a frequency interpretation of
probability and in doing so had denied that could talk about the
probability of a singular event. Von Mises argued that probabilities
could only be defined over "collectives", not singular events. The
example considered was the probability of death. Von Mises had agreed
that we can certainly consider the probability of death before 41 in a
sequence of 40-year-old Englishmen. He said that probability is simply
the limiting frequency of those in the sequence who die before 41. But
von Mises denied that we can say anything about the death before 41 of
a particular Englishman, say Mr Smith. In his words,
"We can say nothing about the probability of death of an individual,
even if we know his condition of life and health in detail. The phrase
'probability of death', when it refers to a single person has no
meaning at all for us." (von Mises, 1928, p. 11)
Popper agrees that idea of taking bets on Mr Smith's demise before he
turns 41 would allow a subjectivist (Bayesian) account of the
probability of the singular event of Mr Smith's death. But Popper was
seeking an objectivist account of probability (recall one of his books
is called "objective knowledge"). So the subjective approach does
not satisfy him.
In 1957 and in 1959 Popper argued as follows. Begin by considering two
dice: one regular (fair) and the other biased so that the probability
of getting a particular face (say the 5) is ¼. Given this setup
consider a collective mostly consisting of throws of the biased die,
with a very occasional toss of the fair die. Since nearly all the
throws in the collective are of the biased die the probability of
getting a '5' must be close to ¼. But let us take one of the rare
tosses of the fair die. Popper asks, 'What is the probability of
getting a '5' on that throw'? According to von Mises it must be
¼ because it is part of a collective where that is the probability of
obtaining a '5'. Popper finds this paradoxical, suggesting that
intuitively it is more reasonable to suggest that the probability of
getting a '5' must be 1/6 for any toss of the fair die.
Popper suggests that the above argument raises a difficulty for a
frequency theorist. The frequency theorist may try to get out of the
difficulty by saying that a sequence of throws of the biased die with
some rare tosses of a fair die interspersed is not a genuine
collective. Popper comments on this adjustment as follows:
"All this means that the frequency theorist is forced to introduce a
modification of his theory - apparently a very slight one. He will
now say that an admissible sequence of events (a reference sequence, a
'collective') must always be a sequence of repeated experiments. Or
more generally, he will say that admissible sequyences of must be
either virtual or actual seuquences which are _characterized by a set
of generating conditions_ - by a set of conditions whose repeated
realization produces the elements of the sequences. [...] yet, if we
look more closely at this apparently slight modification, then we find
that it amounts to a transition from the frequency interpretation to
the propensity interpretation." (Popper, 1959, p. 34)
So the generating conditions are now seen as having the power or
disposition to produce the observed frequencies. Popper notes:
"But this means that we have to visualize the conditions as endowed
with a tendency or disposition or propensity, to produce sequences
whose frequencies are equal to the probabilities; which is precisely
what the propensity interpretation asserts." (p. 35).
[There are a variety of propensity theories of probability, but perhaps
enough has been said to make it clear what a propensity theory of
probability is about.]
(Much of the above is based on Donald Gillies, "Philosophical
Theories of Probability")
.
- Follow-Ups:
- Re: Propensity
- From: Dave Smith
- Re: Propensity
- Prev by Date: Re: Varieties of compatibilism 1 (to be continued)
- Next by Date: Female physica attractiveness
- Previous by thread: Why German women are choosing not to have babies
- Next by thread: Re: Propensity
- Index(es):
Relevant Pages
|
Loading
