Fig.1
The word 'probability' is ambiguously used to refer to two
distinct ideas:
(i) Chances, which are conjectured to be a property of Physical
systems - a property, or propensity, independent of the state of
human knowledge.
(ii) Degrees of belief or betting quotients, which describe the
extent of human confidence in the truth of a claim.
Chancy systems are analysed following the approach of Von Mises and
Popper: Real Physical systems behave in various ways; to describe
some such systems' behaviour, humans devise a model in which a system
has the property of being able to generate, in the long term, an
infinite sequence of outcomes (a collective) with each outcome having
a particular relative frequency of occurrence. This relative
frequency provides the numerical value for a chance. In the short
term no pattern appears in the outcomes.
An outline of the Physics of Poincare, Hopf, and Engel, on arbitrary
functions, in included, to indicate how systems, while fully
determined by laws and initial conditions, come to behave in this
characteristic way.
Subjective degree of belief has no necessary relationship to chance.
It is methodological, concerning what people judge to be a reasonable
extent of confidence in a claim, given the available evidence. The
gulf between chances and reasonable degrees of belief can be bridged
by an Inductive Presupposition, which itself seems to be
unjustifiable, but is in universal use.
This Dual Theory fits unproblematically with all our intuitions
concerning Probability. Traditional problems have arisen as a result
of excessive emphasis either on objective chances, or on subjective
degrees of belief.
{I thank Rom Harré, David Papineau, Brian Skyrms, and John
Welch, for valuable comments on a previous version of this paper}
"Distant stars are moving away from us". "This coin will land
displaying a head". In everyday language, both of these claims are
described as
"probablex'1
(Footnotes are collected at the end of the essay), meaning that
people are not able to establish them as true or false. But, despite
this common feature, they are very different: the first only concerns
a human degree of belief (betting quotient) in a claim, given the
available evidence; it applies to all types of conjectures; this is
epistemic, more subjective. The second concerns both this, and a
conjectured feature of the natural world: 'chance'; this is
ontological, less subjective.
In everyday communication, context and empathy identify which idea we
intend. To emphasise one aspect, as an Objectivist or a Subjectivist,
is, though temptingly simple2, to
oversimplify. Such an attempt is a 'sub-theory'. Consider: "People
have rational degrees of belief in propensities, given the
evidence of relative frequencies". Each of the highlighted
words and phrases is associated with a sub-theory. Since our Dual Theory includes part of each sub-theory, it cannot usefully be
categorised as either 'Subjective' or 'Objective'. It contains both
elements, working harmoniously side by side.
What is the Dual Theory a theory about ? How can it be
assessed? Our evidence - the facts for our theory to explain - is,
firstly, the everyday uses of the word
'probabilityx', and, secondly,
since we are typical users, our intuitions. Our problem is to provide
an organised, summary of these uses - to summarise explicitly and
truthfully the implicit ideas which are guiding the usage.
Though we do not assume that there is one essential concept
present, we do assume that people have some coherent ideas
when they use the word
'probabilityx'.
We therefore assess our theory by its (a) consistency (b) accordance
with our uses and intuitions. This is what is meant by 'analysing',
'unpacking', and 'giving an account' of, a concept.
The challenges that such a theory faces are typically that it cannot
make sense of a particular familiar intuition: say, the intuition
that I have a definite probability of dying in the next year, or the
intuition that a 5 has a probability of 1/6 of being thrown, not say
1/2 (because it can come up either 5 or not-5). The response that the
intuition is mistaken, can only be made with caution
3.
Our primary aim is to find the truth about these uses and intuitions.
It is a reasonable subsidiary aim both to describe the facts of human
use and intuition in a simple unitary system, and to mark the
limits of such simplicity. But it is not reasonable to
insist on such simplicity. Whether it exists, or not, is an empirical
question.
Similarly, it is a reasonable subsidiary aim both to describe the
extent of justification for these human uses and intuitions - and to
mark the limits of such justification. But it is not
reasonable to insist that such justification must always exist
4. Again, whether it does or not is
an empirical question.
The Dual Theory
The word 'probabilityx'
is used to refer to two complementary things, one relatively less
subjective, the other relatively more subjective:
(a) Chance : Some particular Kinds of physical system are
conjectured to have the property (propensity) of chanciness: they
generate outcomes which are very hard for humans to predict in the
short term, yet display steady relative frequencies in the long
term.
This is a matter of Semantics and Ontology - the forming of a
clear conjecture about Nature. It is relatively less subjective.
(b) Degree of belief : Any claim, including ones about chancy
systems, can have a degree of human belief associated with it. This
degree of belief is either personal (based on anything) or consensual
(rational - based on an assessment of the available evidence).
This is a matter of Methodology - of how reliable our
conjectures are, how much we should believe that they are true, given
the evidence. It is relatively more subjective.
These apparently dissociated things have in common only that
they both concern human uncertainty; in (a), because of the nature of
the system, we are uncertain of its short-term outcome; in (b)
because of general lack of evidence, we are uncertain of the truth of
the claim. While this family resemblance fully justifies the everyday
use of one word for both, it has confused Philosophers.
This dual theory is not original. There are repeated
references to it in the literature. I have tried to complete the
model by fitting together available parts - after removing some
inappropriate excrescences, and constructing a few pieces to fill in
the gaps. Nonetheless, to avoid exegetical criticism, I will call it
my theory 5.
Given the considerable number of Philosophers who believe that one or
other of the parts is the whole model, I fear that DT
may not be popular.
I now outline of the two aspects of this theory.
Probabilityx has an objective
aspect. Insurance companies, as Poincaré, for example, wrote,
successfully pay out dividends on the basis of
probabilitiesx; they could continue
to do so, even if further information on the medical conditions of
its clients was provided by unscrupulous doctors - indeed, even if
total evidence was supplied. If probabilityx
ascriptions were entirely subjective, dependent on human
ignorance - and therefore not perceived in Nature by a super-being -
then we would not be able to explain "Why chance obeys laws"
(Poincaré p. 403). Why are we, as non-super-beings, able to
use probabilityx assignments in cases where effects are being
produced by certain kinds of causes, to "successfully foresee, if not
their effects in each case, at least what their effects will be, on
the average"?
Consider, he suggested, the Kinetic theory of gases: we are presently
unable to compute, given initial conditions at a certain time, and
physical laws, how many molecules would hit the side of a box 5
seconds later; we cannot even establish the initial conditions; yet,
oddly, the very complexity of the motions leads us to simple
predictions - which turn out to be true. Even if, with future
technology - perhaps as superbeings - we could do the computation,
and could establish the initial conditions - removing our ignorance -
the predictions based on randomness and equiprobability would still
be correct; chance would still obey laws. The natural system,
consisting of a large number of molecules in a box, has a property,
linked to the success of these predictions, which is independent of
human beings in general, and of their ignorance in particular.
What is this property? Humans have experience of many systems which
appear to have a characteristic Kind of behaviour: their outcomes,
while seeming to occur in approximately constant ratios in the medium
term, hop about unpredictably in the short term. Stimulated by this
experience, humans have developed the concept of a property, which
these systems might have. Without trying to make it absolutely
precise6, we now give guidelines
for a meaning of the concept of
chance1, sufficient, for example,
to help an alien from a non-chancy world to understand what we intend
to mean by the word7.
The conjecture "The
chance1 of an output
of the system being 5 is 1/6" is taken to mean that we are
conjecturing that the system has a characteristic property that
displays8 itself thus:
(i) Long term limiting relative frequency : if, in a
series9 of tests, certain aspects
of the system repeat without
change10 while other aspects
vary11, then the output 5 would
appear with a relative frequency of 1/6 in an infinite series of
tests12
(ii) Short term randomness : The sequence does
not obey any easily recognisable law, or fall into any computable
pattern. There is therefore, for a human whose only evidence is the
previous sequence of outcomes, an inescapable element of doubt
concerning the next outcome13.
Chance1 is thus a
theoretical concept, in the Physicist's sense . Humans can define
any concept they like.
"To what extent does it apply to the external world?" is an
important, separate, question. Compare defining the vis viva
of a moving object as the product of its mass, volume, and speed
cubed. We define the effect that this vis viva has, such that,
when the object's speed relative to a target is within 0.000 000 1%
of 0.5 of the speed of light in vacuo , vis viva, rather than
momentum, is conserved in the collision. So, we have a nice clear
concept. Now we need to obtain evidence to assess whether objects
really have vis viva . And this will be difficult,
because the property has consequences only in circumstances difficult
to test.
This is the situation we are now in with
chance1. We have defined a concept.
We can see immediately that the claim that it applies to a particular
real system will be difficult to test (i) because of the reference to
the sequence being infinite (ii) because the non-existence of a
pattern in a sequence is hard to establish.
This is for later. Our analytical, onto-semantic, task with
respect to chance is complete . We are now at liberty to
use the word, and the idea, freely.
As already explained, the only thing that this methodological,
Epistemological, area has in common with the conjectures concerning
Nature in Element 1 is that both involve uncertainty. Philosophically
this family resemblance is unimportant. The two elements are conceptually unrelated . This is a vital source of
long-standing confusion.
To what extent do we have confidence in a claim C, given indecisive
evidence E (insufficient to establish that C is true or false). It
concerns the extent to which we would bet that C is true - the
extent of our degree of belief in C.
Recalling our meta-methodology, we are aiming to describe, in as simple
a way as possible, the judgements that people make. Then we are
aiming to describe the extent to which these judgements are
justified.
Firstly we distinguish reasonable degree of belief
mB and individual degree of
belief iB. The former is
characterised by consensus agreement; the word 'reasonable' does not
imply that the judgement is going to be justified - it is merely a
convenient label. The latter is any personal degree of belief,
regardless of what the consensus may judge.
In this next section we discuss the best available principled
description of human reasonable degrees of belief, which is that
according to Bayes' theorem.
Degree of belief can be roughly classified as from 0 (no belief) to
1(total confidence). mB is part of
a triadic relationship between {claim, evidence, reasonable degree of
belief}. Values of mB are
intuitive, part of everyday life and scientific method; judging by
the failure of many attempts, we do not think that there are simple
rules to summarise them, other than Bayes' Theorem . This says
that our reasonable degree of belief in a claim, given the evidence
and background knowledge, is increased by:
(BTi) increases in our degree of belief in the claim, given
background knowledge
(BTii) decreases in our degree of belief in the evidence, given
background knowledge
(BTiii) increases in our degree of belief in the evidence, given the
theory and background knowledge.
As Howson and Urbach show convincingly in their (1993), this theorem
satisfactorily summarises the facts of human use and intuitions
concerning the triad {claim, evidence, reasonable degree of belief}.
In particular, for example, it summarises the vital role of novel
fact prediction , as a human method in the search for the truth
about Nature.
Howson and Urbach also explain, with unusual clarity and firmness,
that the theorem does not justify human
behaviour14. Its only normative
force, they insist, is that a person who denies a consequence of the
theorem, but insists that he is using 'degree of belief' (or,
loosely, 'probability') in the usual way, is guilty of inconsistency.
In other words, the theorem captures a key aspect of the standard use
of these words.
The role of chance in degree of belief : This is the first of
the confusing links between the two elements of our dual theory. Our
phrasing of Bayes' Theorem was odd, because we avoided the word
'probabilityx'. BTiii would usually
be expressed as "the probability of the evidence, given the theory
and background knowledge". But, given our theory, this means either
'degree of belief' or 'chance'. Which is it?
Consider the successful prediction by Fresnel, using his new
wave-theory of light, of the spot of light in the middle of the
shadow of a small object. This novel fact prediction increased
Physicists' degree of belief in the wave theory, following their
intuition that the chance of a random theory with no truth in it
successfully predicting this observation was very small.
Physicists were comparing the reasonable degree of belief in two
meta-theories: ( MTt) Fresnel's
theory contained some truth; (MTf)
Fresnel's theory is totally false (entirely a human fiction). The
degree of belief in each of these meta-theories, given background
knowledge, is equal (say), and the degree of belief in the novel
fact, similarly, is equal. But Physicists assessed the chance of the
dot occurring, given MTf, as very
small. This leads to a very low degree of belief that the dot would
have occurred, given MTf. This then
implies, inverting according to Bayes' theorem, a very low value for
the degree of belief in MTf, given
the dot occurring. In other words, it is not reasonable to believe
that Fresnel' theory is entirely a human fiction, given this
evidence.
Summarising this use of Bayes' theorem: People are inclined not to
believe in the truth of a theory if events which, according to the
theory, have a very low chance of occurrence, and which therefore
they would tend not to believe would be observed, have been
observed.
This, we suggest, is well described by Bayes' theorem . It is,
as I have indicated, an aspect of Bayes' theorem which is relevant to
the indirect testing of any theory. The importance of this
fact becomes apparent when we consider the potentially confusing
situation where the theory to be tested is a chance; where Bayes'
theorem describes a reasonable degree of belief in the truth of the
conjecture that a Physical system possesses a chancy
quality.
This almost self-referential situation, combined with refusal to
accept that justification may be impossible, combined with a tendency
to use 'probabilityx' to mean a
mixture of objective chance and subjective degree of belief, has been
the cause of some Philosophical confusion.
The Inductive Presupposition
In this section we consider the extent of justification for
Bayes' theorem. Every second of their lives, humans make an Inductive
Presupposition IP. As generalising creatures, they instinctively jump
to general conclusions from particular experiences. They presume -
they believe - that their spatio-temporally local experiences have
been, and will continue to be, typical of the Universe. They presume
that Nature will not use their experiences to mislead them. Sceptical
Philosophers have long suspected that they lack justification for
this presupposition - for this belief. As a key step in our analysis,
these located sceptical doubts are now quarantined. We are not
implying that they are solved; we are merely separating our
variables. If the reader thinks that they are solved, good - she
should insert her solution before the presupposition, and move on. If
the reader, like us, thinks that they are not solved, then she should
note that an unjustified presupposition has been made, and move on.
Either way, the important point is that she should note the
presupposition and move on. So we now define 'reasonable/IP belief'
to mean a belief that is reasonable, conditional on the universally
accepted Inductive Presupposition/IP.
Predicting, and Testing - the route to Bayes' theorem
Degrees of belief in consequences, given chances (Predicting):
Suppose that we accept that the chance of event e occurring is 0.000
1. Then our reasonable/IP degree of belief in the event, given this
chance, is 0.000 1. Degrees of belief in chances, given consequences
(Testing): Inversely, suppose that we accept that an event e does
occur. Then our reasonable/IP degree of belief in the chance of it
occurring, given that it has occurred, is high. If an event occurs,
then our reasonable/IP degree of belief in a theory which claims that
the chance of this event occurring is extremely low, is extremely low
(unless we have very strong other reasons for believing the
theory).
To what extent are these two moves justified? Isn't a chance
hypothesis consistent with any finite sequence of outcomes, however
long? To what extent is it fair to bet 1:10 000 on e occurring in the
next test, conditional on the chance of e being 1/10 000? It is not
fair; it is not justified. But it is fair/IP and justified/IP. If our
experiences are typical, a fair sample, of how Nature behaves, then:
(i) we can reasonably/IP believe that we will not observe the
occurrence of an event which has a very low chance of occurring
(Predicting observed events). (ii) we can reasonably/IP disbelieve
theories whose truth would require, unreasonably/IP, that our
experience was untypical (Testing theories)).
Substituting the first link into the second, we conclude that if an
event occurs, then our reasonable/IP degree of belief in a theory
according to which our reasonable/IP degree of belief in the
occurrence of this event would have been extremely low, is extremely
low (unless we have very strong other reasons for believing the
theory). The two reasonable/IP degrees of belief are correlated. In
notation:
mB(h/e) is proportional to
mB(e/h) X
mB(h)
This discussion need not be prolonged. It is heading, qualitatively,
towards Bayes' theorem, which we have already agreed to be an elegant
summary of a cluster of human intuitions as to what is reasonable/IP
to believe. Howson and Urbach's admirable book provides many
examples.
But why, it may be asked, did we go to this trouble, when we were
already using Bayes' theorem at the beginning? There are two reasons:
(i) If 'probability' is ambiguous, then the calculus of
probabilityx, and statements within
it such as Bayes' theorem, will tend to be tarnished by confusions
engendered by a unitary interpretation of P(x). (ii) We needed to
establish not only a principled description, but also the extent of
justification, for human behaviour. The calculus, including Bayes'
theorem, is a principled summary description of the use of the word
'probable' in everyday language - a summary of the idea (concept), as
used and approved by the community of users. It summarises the use
which is regarded as reasonable, that which is accepted by the
consensus. The limit of its normative force is to impose consistency:
if a user claims to be using the word in this accepted way, and then
does not accept some result which follows within the calculus, he can
be accused of inconsistency (as Howson and Urbach emphasise). But it
provides no justification for these uses - nor does it aim to do so.
Indeed, by following everyday consensus usage, it precisely
replicates the human attitude to Philosophical doubts concerning
Induction - it ignores them. Mathematicians, Statisticians, and those
who use their results, instinctively ignore such doubts. Our
Philosophical aim, by contrast, required us to (i) separate the less
subjective from the more subjective aspects of
probabilityx, and (ii) assess the
extent of justification for the judgements made in its name. In these
tasks we could receive no assistance from the calculus, nor from
Bayes' theorem. However, our aim also required us to capture, to
reconstruct in a principled form, human intuitions. Since these are
nicely summarised by Bayes' theorem, it was essential that we were
able to reconstruct it.
We now turn to the classic hurdles which loom up, ready to trip a
theory of probabilityx.
In the rest of this paper I indicate how this DT jumps
the following hurdles:
(i) Are conjectures concerning
chances1 empirical? {If no definite
testable consequences can be derived from them, nor any evidence
prove or disprove them, then they are metaphysical, and have no place
in positive science}
(ii) Can we explain the application of probabilities to single
events?
(iii) Can we explain how chances have arisen, on the supposition that
Nature is deterministic? Can we explain how some systems lead to
disorder, and then back to some kind of statistical order?
(iv) Can we explain what the
chance1 that we associate with a
system amounts to, other than the infinite sequence ratio? (Have we
explained what a chance is - in the world?}
(v) Can we explain how chances seem to vary, depending on the choice
of outcome space? {If we regard the die as having 6 outcomes, then
the chance of getting 5 is 1/6; if we regard it as having 2 outcomes
- 5 or not-5 - then the chance of getting 5 is 1/2}
(vi) Can we account for conditional probabilities?
(vii) To what extent do we have evidence that any real systems are
approximately chancy?
(viii) Suppose that a die has been thrown, at t = 0, and a 5 has just
been obtained. What was the probability of this event occurring? {Was
it, for example, 1/6, or 1?}
(ix) Can we explain how the system, and indeed the outcome, is to be
specified? After all, if the system is specified too precisely, there
may be no variation in the outcome, while if the outcome is specified
too precisely, every one will be unique. Doesn't ambiguity over the
Unique Experimental Protocol - the specification of the system - make
the objective probability unacceptably variable, for a quality that
is supposed to exist in the external world?
(x) Wouldn't it be preferable to stick to reasonably definite,
testable, things like degrees of belief (in the form of betting
quotients, and utility), rather than conjecturing the real existence
of peculiar qualities of systems, which are not positively
testable?
(xi) Does ignorance lead to equiprobability?
(xii) What happens if we interpolate two throws of a fair die into a
sequence of throws of a heavily loaded one?
I hope that this list includes your favourite hurdles for a theory of
probabilityx. I will tackle them in
the order that they would perhaps have occurred to you - an order
therefore more pedagogical than logical. In the process I will repeat
and develop aspects of the theory.
To what extent do we have evidence that there are some such
systems, at least approximately, in Nature? To what extent can we
establish values for the chances in these systems?
Since this concerns how conjectured chances are tested, it is a
methodological question.
What methods are used to test particular conjectures, such as
"This die has 6 sides"? This is direct testing ; observation
confirms or disconfirms it. This case is unfortunately not relevant
to us.
What about general and theoretical conjectures? This is
indirect testing ; testable consequences are deduced from
them. Verifying these does not, however, establish the conjecture as
true, since many other conjectures could have had the same
consequences. By observation of human behaviour (intuitions) we have
already discovered that humans make the completely unjustified
Inductive Presupposition (IP ): If one theory T1
implies that a consequence C is very likely to occur, and T2 implies
that C is very unlikely to occur, and then C is observed, then T1 is
more likely to be true than T2. This is, of course, entangled with
Induction; it is the presupposition that what we happen to observe is
a fair sample of the consequences of the true general laws and
theories; it is the presupposition that Nature does not deceive us,
when we collect our measly fragments of information from her.
It is, of course, the presupposition that we identified before, in
our discussion of Bayes' theorem. But at that stage we were merely
noting its presence in a description of human intuitions concerning
the triad {claim, evidence, reasonable degree of belief}, as it
applied to all claims. Now we apply it to claims concerning
chances.
Conjectures concerning chances are made empirically testable by the
application of Cournot's Rule (see, for example, Gillies (1973)).
Criticism of this rule is misconceived. His rule is not a part of the
Ontology or Semantics of chance; it is not a part of the less
subjective element 1at all; it is part of element 2 -
the methodology. Consider a familiar example: Our conjecture is C1: a
die has a chance of 1/6 of coming up 5. We obtain evidence E1: we
throw the die 6000 times, and get only 2 5s. We now have a claim, and
some evidence. The evidence is a logical consequence of the
hypothesis - that being an inevitable feature of chance hypotheses.
Now humans use Cournot's rule, which is that, roughly, very small
probabilities are impossibilities. Our form of the rule is IP: "Do
not believe in the truth of theories which imply that events you have
observed have a very small chance of occurring". If our conjecture C1
is true, then E1 had a very low chance of occurring. We conclude that
we should not believe that C1 is true.
Thus our conjecture concerning a chance is indeed empirical, and is
tested in a familiar-sounding
way15.
Have we justified this procedure? No. We judge that the extent of
justification is zero16. This is an
interesting fact about human indirect testing of claims. Can we
define what counts as "very small". No. That is another interesting
observation, concerning the apparent roughness of this human
methodology.
Our unpleasant conclusion is that our best model of chancy systems
produces consequences which are compatible with any finite observed
sequence of outcomes. Chancy conjectures have no justifiable
empirical significance. But this is not a scandal; they merely 'join
the club' of other hypotheses which, being too distant from direct
testing, suffer from the problem of 'Inference To The Best
Explanation'. Human intuition carelessly leaps the logical gap.
The reader should find this unsatisfactory. Of course it is. But it
is the truth about our human situation - our reach exceeds our
grasp.
Following Von Mises, and Howson, (and not, for example, Miller
(1994)) we suggest that single events, if not regarded as part of a
collective, do not have chances1
associated with them. I can coherently, of course, express a degree
of belief that I, as an individual, am going to die in the next year
- and a fortiori I can express a probabilityx that I will die.
I can coherently conjecture a chance of dying if I consider
myself as a person, or as a man, or as a man aged 48, or as a man
aged 48 who takes a bit of exercise, but if I insist on cutting
adrift from all collectives, the sentence "I have a high probability
of dying in the next year" can only express a degree of belief.
Von Mises' chance1 cannot,
consistent with its meaning, apply to events at all, multiple or
single. A single event is generated by the system; a collective of
events is, hypothetically, generated by the system - it makes no
difference; the property chance1 is
a Physical quality of the system - the property of tending to
generate collectives. In other words, the question: "What is the
chance1, (as a 40 year-old person
who has just signed on to the Life Insurance Company) of the event
'Me dying in the next 10 years'?" is a misposed question. This is
unsurprising: "What is the weight1
of his troubles?", where 'weight1' is as defined in Physics, is
similarly misposed.
In this case, we can continue to talk of our degree of belief that we
will die in the next 10 years; we may even have a betting quotient
associated with the degree of belief; but this has nothing to do
with chance1
.
In other words, if people ask: "What is the
probabilityx
associated with the event 'Obtaining 5 on the throw of this die
at t +1'?, they are making either a concealed reference
to the chance1
associated with the system generating a 5, or a reference to
degree of belief.
Is there any reason why we might want
chance1 to be a property of events,
or indeed of a single event, rather than - or as well as - a Physical
property of a single system?
(a) "Some patterns of everyday speech seem to have the form of an
assignation of a probabilityx to an
event" . This is trivial. It is typically a degree of belief.
(b) "We would like chancex to be a
single-case propensity". This is meaningless. A propensity is a
property of a Physical system.
We here summarise the approach of Arbitrary Functions, due to
Poincaré. It provides a model for how the phenomena we call
'chances1 ' could arise naturally
in a deterministic world.
The Experiment
On the bench in front of us is a gas container, connected to an
electronic device, on which is a blank display. When we press the
button beside the display, a number appears. We press it a couple of
times; numbers between 1 and 6 appear. They show no immediately
obvious pattern; they hop about. We record them for 360 presses
(tests); each appears about 1/6 of the
times17. We record 360 000 tests;
each appears very nearly 1/6 of the times . This is an interesting
phenomenon. We cannot predict individual outputs, despite our best
efforts to find some pattern to the sequences. But we seem to have a
physical law that we can use to predict ratios for large numbers of
outputs.
We now decide to study the system on the bench. We hope to devise a
Physical model of the system, to see if our model might display, in
the short and long term, the characteristic behaviour.
The computer simulation of a model of the gas in the
container
We find that the gas container has a small pressure sensor inside it.
This generates a voltage V , proportional to the
pressure detected. When we press the button, the device samples this
voltage. If it has value a the device outputs a 1 to
the display. If the voltage is a + e ,
where e is a very small value, then it will output a 2;
if to a + 2e , as a 3; and so on, in a
cycling sequence18. In other words,
the device's output is very sensitive to small variations in
its input. Because different numbers of molecules hit the sensor per
second, the pressure on the sensor varies, and therefore the voltage
varies, giving outputs which hop about.
Since the voltage is the key variable which links the two parts of
the system, we call it the 'Poincaré variable'.
Which part of the system is responsible for the short-term
unpatterned, yet long-term patterned, variation in output? If it is
not the electronic processor (the secondary system ), it must
be the gas and sensor (the primary system ). The molecules in
the box, starting with some initial conditions, and governed by, say,
deterministic laws, display behaviour over time - including the
entire sequence of outputs - which is
determined19. Nonetheless, the
pressure on the sensor is varying in a way which, though hard to
predict, has some kind of long-term pattern in it.
The gas contains about 1027
molecules, so we are unable, as humans in 1997, to
calculate what will happen. We cannot even establish where all
the molecules are, and what their speeds are, at t = 0,
far less calculate, using classical mechanics, where they will all be
at t = 5, how many of them will hit the sensor, how
hard, and therefore what the pressure on the sensor will be. We are
reduced to argument, and computer simulation, to work out what is
happening.
Suppose, following Poincaré, that our first simulation is very
simple - too simple. At t = 0, all the molecules are
projected in the +x direction. They are in a perfect
cube. with idealised smooth walls at which molecules rebound, with
i equalling r . This simulation produces
outcomes which time-repeat, giving a cyclic pattern to the outputs
which we have not observed. It gives no sign of the characteristic
behaviour we are seeking.
This failure is not surprising. Our idealisation left out
essential aspects of the situation. The real container is not
a perfect regular shape. The molecules will change direction after
their first collision. They will hit a rough wall - at the molecular
level - and not obey the law of reflection. An infinitesimal
variation in their original path will cause a finite change in their
succeeding velocity. Infinitesimal further changes, due to collisions
with other molecules, will then produce further large changes in
their direction.
Sometimes, as when Galileo initially left out air resistance, an
idealised model can still give very accurate predictions, covering
all the main aspects of the phenomenon - leaving only details to be
tidied up. But other times, especially when positive feedback, or
non-linear equations, govern aspects of the system, leaving one
apparently small factor out of the model can lead to inability to
derive major aspects of the behaviour of the real
system20.
We run our second, less idealised, simulation, on a more powerful
computer, with more realistic walls - consisting of lots of tiny
bumps and dips - and realistic interactions between the molecules. We
find that if all the molecules start off in one direction, at one
speed, then, after a certain number of collisions, the order is lost.
Very soon "their final distribution has no longer any relation to
their original distribution" (p.401). If the simulation is run on,
with 1000 000 molecules, we find that a simulated sensor in the
corner indicates a characteristic kind of variation in number of hits
- a variation which recurs however we set the molecules off. The
immensely complex simulated behaviour of the more realistic situation
has led, in one respect, to a law-like simplicity.
The simplicity is this. The simulated sensor repeatedly records the
number of molecules that hit it in successive 1 ms intervals. In the
first interval it records, say, 1000 atoms. In the next, it records
890. After, say, 100 s of recording these numbers, it processes its
data, to find the frequency of occurrence of intervals in which
particular numbers of atoms arrived. It finds that between 986 and
995 atoms arrived, in 100 of these intervals; between 996 and 1005
atoms arrived, in 102 intervals; between 1006 and 1015 atoms arrived,
in 101 intervals. In other words, the number of times that similar
numbers of molecules arrived, is very similar. The larger the number
of molecules we simulate, the more accurately this holds true.
Suppose that we cannot yet explain, using only our objective
theoretical description of the situation, why there is this
characteristic similarity in the number of times that one number of
hits are recorded, and the number of times that very similar numbers
of hits are recorded. This present failure does not imply that the
characteristic similarity is not objective ; it just implies
that we have not yet managed to tease out decisive predictions,
explanations, of every aspect of the behaviour of the system,
by methods other than modelling. Why should we have done? Computer
modelling is regarded by Physicists as a fully acceptable substitute
for solving equations, in cases where the latter has proved
impossible.
In our simulation, the ratio of occurrence of pressures in one
interval, which we could label b , has come out very
similar to the ratio of occurrence of pressures in a closely
neighbouring interval21, which we
label b + f . The longer we run the
simulation, and the more particles we put into the box, the more
accurately this result is obtained. We call this property of the
pressure variable, 'Poincaré
variation'22 (Poincaré
pp.403-4).
Comparing the simulation to the real world, we find evidence, by
direct measurement of the pressure on a sensor in a real gas, that
this variation is indeed occurring. Now we recall that the sensor
converts this varying pressure to a varying voltage. Therefore the
voltage will also vary such that if a occurs a certain
proportion of the total number of times in a long sequence, then
a + e will occur the same proportion of
times, as will a + 2e . Similarly, if
a + 5e occurs a certain proportion of the
times in a long sequence; we know that a +
6e occurs the same number of times.
Suppose that a , a + 5e ,
a + 10e , and so on, cause the device to
display 1. a + e , a +
6e , a + 11e , and so on,
cause the device to display 2. And so on.
If therefore the electronic device continually recorded all the
voltages received, it would end up with equal numbers of each
output, from 1 to 623. We are
closing in on the appearance of equichance for the six outcomes.
Suppose instead that the device does not produce an output until the
button is pressed. If a is detected, then 1 is
displayed as output. If a + e is
detected, or a + 6e , then 2 is output.
We now simulate, on our computer, different patterns of button
pressing. We are facing the problem of sampling , in various
ways, without appealing to randomness. We first try patterns where
the button is pressed at regular intervals. We try patterns where the
intervals increase and decrease according to periodic functions. We
try every ordered pattern we can. We find that we always get the
ratios 1/6 for output 1, 2, etc.. , just as we did in the real
experiment. We are aware of the possibility that a sequence of
presses could give a quite different ratio - we could get 1 every
time the button is pressed. But every case we try gives these
ratios. We can simulate the actual times of an experimental run. The
functions that we have tried, cover a large number of times of
pressing. We find that, with our simulation, only once in, say, 106
trials do the ratios not come out to 1/6.
Returning to the real world, this fits with the human case, where
people, whether trying to follow simple patterns of pressing (every 1
s), following highly complex patterns, or pressing when they feel
like doing so, get the ratios to come out to 1/6 about the same
number of times (if they are patient enough).
'Sampling every 1 s' is an easy sampling pattern to model. 'Pressing
when I feel like it' is a hard pattern to model. If we suppose that
the human being does not introduce some new kind of randomness into
the world - if we suppose that we are a complex, Natural, electrical
system24 - we could roughly
simulate the working of the brain. Viewed as a complex system
producing an output at irregular intervals, we could model it as a
second complete system of the above kind. Linked electrical sections
of the brain (neurons) give a chaotic variation in a voltage at one
point in the brain, acting as a Poincaré variable. Each time
this variable, say, rises above 0.2 mV, the person presses the
button; he also gets an inclination ("I feel like pressing it now")
to press the button.
So, in a sequence very difficult to predict, the first system has
its button pressed25. Run after
run, the computer simulation gives output ratios of 1/6 for each
output.
We can also link the sampling to other parts of the natural world
which seem to display this characteristic feature. We could link it
to another identical system, so arranged that when more than 1000
atoms hit the second sensor in 1 s, the second system outputs the
command to push the button on the first. Alternatively, we could
press the button if a Geiger counter detects more than 100 counts
from a radioactive Cobalt-60 sample in the preceding second. One part
of the Physical world, generating a certain kind of varying variable
and linked to a sensitive processor, when it interacts with another
such part, tends to produce, 99.9999% of the times, a characteristic
patterned output.
GENERAL PHYSICAL FEATURES OF SITUATIONS IN WHICH CHANCES
ARISE
The computer simulation shows that certain systems, even when
micro-deterministic, could develop the kind of characteristic
behaviour which we have agreed to call 'chancy'. We conjecture that
these systems include not just the real case discussed above, but
coin tossing, roulette, and other classic systems.
In each such system:
(i) There is a conjecturally deterministic primary system .
Typically because it contains large numbers of moving independent
molecules, governed by processes including positive feedback, it
generates a physical quantity, the Poincaré variable, which
displays characteristic Poincaré variation (the ratio of
occurrence of one small interval of values is very similar to the
ratio of occurrence of the adjoining small interval, over an
infinitely long sequence of tests). This variable is the input to the
next system.
(ii) The Poincaré variable is fed into the secondary
system , which is a processor , which processes the
varying input in a characteristic way, being sensitive to small
changes in the Poincaré variable. It then produces a selection
of specific outputs .
The vital feature of this description is that objective features of
the world have merely led to further objective features. No aspect of
the above involves reference to the limitations of human
knowledge concerning the outcomes of the systems.
This completes our theory of chance1
- objective
probabilityx (see diagram
below).
The simulation model shows that micro-randomness can emerge
from a micro-deterministic system. This fills in the step before
Poincaré (see also Knitchine (quoted in Gillies)), who argued
on from micro-randomness to
macro-randomness26. We do not need
to answer the objection that you can'tx get chances out unless
you put them in. Unless the critic can explain in what sense he means
'can't', we can merely point to the above passage, and indicate that
it shows that chances do , and therefore can , emerge
from deterministic systems. This short-circuits any attempt at a
conceptual argument that this is somehow impossible (perhaps because
of the meanings of the words involved).
We are not claiming that personal
probabilityx, reasonable degree of
belief or betting quotient, emerge from the Physics of the external
world; these concepts are person-relative, arising when we face
events we have difficulty in predicting. We are claiming that the
characteristic property of systems in the world, which leads them to
produce such events, could be a predictable consequence of
deterministic Physics. We suggest that our simulation presents the
reader with an example defying his faith. To just state that this
cannot be happening is like stating that a rocket cannot
accelerate in space, because it has nothing to push against, even
when presented with evidence of a rocket accelerating.
Where is the fault in the simulation? We suggest that it provides
evidence that chancy1 behaviour can
be generated from entirely deterministic micro-behaviour.
Gettingx
probabilityx is vague. We cannot
expect to get the concept of chance1
from a simulation; that is a category mistake. We can
try to get chancy1behaviour
from a simulation.
Have we made some assumption about the chances of the presses? Not
necessarily. The system, following deterministic laws, generates a
sequence of numbers if pressed every second . This pressing is
not chancy - it could be done by a mechanical clock. No part of the
system, nor of the pressing, is smuggling chance or randomness into
the system.
But couldn't it happen, by chance, that the pressing gives a 5 every
time? Certainly, but this does not imply that we have failed to
devise a chancy system; this is a typical property of a chancy
system. Once again we are confusing the ontology with the
methodology. The chanciness of the system is perfectly compatible
with a particular observed finite sequence being apparently not
chancy.
Is the preceding rather scientific section on molecules and
chaos actually necessary to the view?
Maybe not. Suppose that the world had turned out, as far as we can
tell, to be tychistic. Suppose, in other words, that the die tossing,
the gas experiment, and so on, all behaved as they do on Earth, but
that our further investigations of the structure of these systems did
not unravel any deterministic laws. We could see that there were 6
outcomes, but beyond this we could not get. Either there seemed to be
no lower-observability parts of the systems, or, if there
were, they seemed to be all vague, loosely defined, and
indeterministic. Perhaps we had reason to suspect that human
ingenuity would be permanently blocked from discovering what, if
anything, was actually going on.
We could still propose that the system was objectively
chancy1, and that the value of the
chance property attached to each outcome was 1/6. There need be no
suggestion that the chance was a measure of human degree of belief,
or betting quotient. We would be in the position of Galileo saying
that a sheet has some colour property, which causes it to produce in
us the sensation of whiteness; this is an objective property - but he
had no idea what was causing or underpinning it. The chance would be
a property of the system, just like any other property.
If we later found that the system actually had some hidden
variables , whose variation was causing the
chance1 property (as shown by
computer models), then this would not alter our description of the
system as possessing objective chanciness. And if instead we did not
find this, then we could reasonably stick with the view that, say,
Nature is essentially tychistic, such that randomness and
chance1 outcomes are a fundamental
aspect of macroscopic Natural systems because chanciness is built
into them (what could be called 'genuine indeterministic
chance').
Does our computer model approach fit with the basic theory of
statistical mechanics and thermodynamics?
(See e.g.. Sklar Physics and Chance )
(i) Will our model explain how the gas reaches the equilibrium state
in which a gas in the actual world mostly is?
(ii) Will our gas display Poincaré recurrence? At some stage
it will reach exactly the state it was in at the stage when we
released all the molecules. Does this matter? Does this disobey the
law that entropy increases?
Our gas will also display time reversal symmetry, such that it could
run backwards in time, consistent with physical laws. In other words,
we do not establish an arrow for time. Does this matter?
Our inclination at present is to suggest that the laws of
thermodynamics are themselves probabilistic, in the sense that they
are not precisely true. What is our evidence that all systems
always, to any degree of precision, tend to increase in entropy? We
see no reason why a gas in a container should not start with its
molecules all along one side, about to travel in the x-direction at a
constant speed (low entropy), proceed, as the simulation indicates,
rapidly to a state of disorganisation (increasing entropy), continue
for some variable length of time in various disorganised states, and
then, for an instant, return to its initial state (low entropy), and
then the cycle begin again. Do have any evidence that this does not
happen? It seems a reasonable sequence of events.
Consider the classic experiment of James Joule, based on one by
Gay-Lussac: a gas is in one of two containers, linked by a tube, but
separated by a valve. The second container is evacuated. When the
valve is opened, Joule found that the gas tended to fill both
containers, though there was no change in the overall energy of the
system. Entropy has increased, and the Second Law implies that the
process is not reversible. But again, we could harmlessly suppose
that the law is a version of our Inductive Doubt quaranting - it is
saying that we do not tend to observe very
low-chance1 events. We could say
this, and still say that that, according to our theories, they
will occur.
Poincaré recurrence for a gas in our container will
occur, in our simulation, after a certain amount of time. But if,
more realistically, we change some tiny aspect of the container, or
motion of some particles, before this happens, then the recurrence is
spoiled. Such a change every hour will ensure that recurrence never
occurs. In the real world, either as a result of the bombardment, or
because of external changes, this is just what happens. The tiniest
change in any aspect of the external Universe will produce
this effect, because of, for example, the change in the gravitational
force on the molecules. We conclude that the periodic recurrence of
the behaviour of a gas in a container only occurs in an
oversimplified model, not in the real world.
Some systems, governed by positive feedback of small changes, tend to
behave in a characteristic way. The system's tendency to behave in
this way could be described as a property of a characteristic
Kind (associated with a ratio) - the system has the power, or
property, or quality, of tending, in an infinite series of
independent tests, to generate a limiting frequency, and randomness.
This objectivex aspect of the system, which causes it to behave in
this characteristic way, is referred to, in everyday English, as
'chancex' ('games of chance'), or
as 'probabilistic'. We will call such systems
'chancy1' (this being an ugly
version of the more familiar 'probabilistic'.)
This aspect is objectivex, in that
its existence has nothing to do with degrees of belief. The
characteristic property, used to identify the Kind, is that:
(i) the outputs in the short term are hard for humans to predict -
they seem to be hop about lawlessly - yet:
(ii) the outputs in the long terms appear to be governed by lawlike
ratios we can associate with them.
To say that a system has a characteristic property , is to say
no more than "They do certain things, make certain things happen, in
certain situations". When it is not in these situations,
although it is not doing these things, it retains the ability
to do them, when its situation changes. How we describe this is up to
us. The particular choice of language of 'qualities' has always
created difficulties. Some systems behave in a certain characteristic
way in some situations. To say that this is because they
possess a certain property, makes us seem to be explaining the
behaviour, when we are doing no more than locating it. We are
just finding a different way of describing what the system can
do.
The characteristic short and long-term pattern in certain events, and
the tendency of certain systems to produce this pattern, exist and
have been explained. There is no further answer to the question "What
is chance1 ?".
'Chance1 outcomes', which is the
highest-observability objective aspect of chance, are the aspect of
the world which is displayed by the outputs of a system of the Kind
described above. A superbeing would regard our six outcomes as
similar in a characteristic way. She would suggest the coining of the
term 'chance1' to refer to the
occurrence of a particular one of the six outcomes. In general, she
would suggest the use of the word, objectively, to describe a
particular outcome of a system which, with a very small change in its
initial conditions - where such changes are occurring - would have
led to another, considerably different, outcome. She classifies the
outcome as 'the result of chance1';
the 5 is obtained 'by chance'.
She also suggests the use of the word to refer to the feature of the
situation that, over finite repeated outcomes, using the
Improbability Presupposition, a particular outcome will tend to be
recorded a certain specific proportion of the times (the limiting
frequency); the asymptotic ratio resulting from an infinite sequence
of tests gives the true chance1.
Thus she now refers to an outcome which has a chance1 of
occurring of 0.527.
The superbeing has identified a particular interesting kind of
situation, characteristic of the present state of the
Earth28, which she calls
'chancy1 '. She would be able to
assign the same chances1 to the outcomes of trials of the die in a
situation, even though she knows what all the outcomes are going to
be.
Could we have done without chances, and make do with relative
frequencies? We cannot, because the relative frequencies are not the
property, they are the result of the property, that the system
has. True, when we refer to the chance1
of a certain output being 0.5, we are referring to the
relative frequency of its appearance in a putative infinite series of
independent tests. This series cannot be undertaken. But when we
refer to the system as 'chancy1',
we are referring to a property that the system has, the
property of generating outputs which hop about in the short term, yet
obey certain relative frequency laws. The property is not a
relative frequency; it is about relative frequencies that will
ensue, if tested. The relative frequency is the outcome of the
property displayed under test.
The chance in a coin-tossing system is not a property of the coin, it
is a property of the system, including, for example, the skill of the
person doing the tossing (a trained person could learn the skill to
control the toss so as always to throw a head). This is a
non-localised relational property .
Finally, therefore, there is no extra property called
'chance', in addition to the 'power to generate limiting relative
frequencies in infinite sequences of outcomes of
tests'29.
If we regard the die as having 6 outcomes, then the chance of getting
5 is 1/6; but if we regard it as having 2 outcomes - 5 or not-5 -
then the chance of getting 5 is 1/2.
This hurdle derives from the classical ignorance theory of
probability, in which outcomes are assigned equiprobability. In
Dual Theory the chances derive from the nature of the system,
and do not display this alarming variability. Suppose that, after a
brief look at the system, we conjecture that C1: the chance of
getting a 5 is 0.5. We then find that, after the machine has
displayed 6000 numbers, only 1010 are 5s. Using Cournot's Rule, this
disproves the hypothesis C1.
Chances are not relative to evidence; they are inhuman; they do
not change as a result of changes in human background knowledge.
There is no such thing as a conditional
chance1.
However, a human conjecture as to the value of this
chance1 does change, as a result of
changes in evidence. And the human degree of belief that the value of
the chance1 is true varies with the
extent of evidence. Thus the degree of belief in a particular outcome
is, for two reasons, relative to evidence, and human. There is
conditional degree of belief.
(a) Outcome evidence
With the help of our concept, humans conjecturally classify
some systems as examples of the Natural Kind called
'chancy1 ', on the basis of
evidence of behaviour, and of structure. This conjecture cannot be
proved true, just like many conjectures in Physics . Its claim is
low-observability, not in the sense that we are conjecturing the
existence of a low-observability entity, but in the sense that the
truth of a universal law is low-observability, because it has an
infinite number of consequences. We cannot prove beyond doubt that a
chance is present , because, in claiming that it is, we are making a
claim concerning an infinite sequence of outcomes.
Owners of casinos base their livelihood on the conjecture that their
games of roulette, poker, and blackjack, are systems of this Kind -
systems possessing the chance
property30.
More precisely, they base their livelihood on this conjecture,
combined with an unjustified presupposition, or prejudice - the
prejudice that in the finite sequence of outcomes, generated
by a week of play, the ratios of outcomes will be close to those
associated with the conjectured chance property, and the prejudice
that the finite sequence will display no pattern
humanly-computable by place
selection31. If they are reflective
owners, they will be aware that this presupposition could be proved
false, for a day, a week, or a year; in this event, they will go out
of business. But they presume that the finite sequence will be
a fair sample of the infinite one. By making this presupposition they
are Quarantining Inductive Doubt, as all humans do in their
everyday life.
They therefore presume that the finite sequence obtained each week in
their casino will be typical , that it will be a fair
sample of the infinite sequence, that Nature will not mislead,
and deceive, them.
They accept that the ratiof in the
finite sequence will not be exactly the
ratioi of outcomes predicted for
the infinite sequence. But they presume that the
ratiof will only differ from the
ratioi by an amount up to that one
would expect, if the sequence had been selected by chance from all
possible sequences, and that the chance of selecting that one was
more than, say, 1 %. In other words, they are presupposing that a
sequence-event whose chance of occurring (in a
human-selecting-sequence system) is less than 1/100 will not have
occurred when they were sampling. In brief, they presume that
events with a very low chance of occurring did not occur when they
were sampling . This, as we have seen, is Cournot's Rule.
(b) Structural evidence
Some features of a system may lead us to conjecture that it
will display chancy behaviour, even without outcome evidence. These
features are those, for example, displayed by roulette, cards, coins,
and dice. By studying the die-human-hand-table system, we conjecture
that it has, say, 6 outcomes. We also conjecture that, if the hand is
more than about 1 m above the table, then very small variations in
the initial position and velocity of the die, will lead to
considerable variation in the outcome. We hence conjecture that no
pattern will emerge in the outcomes. We also conjecture that, in the
long run, each outcome will occur 1/6th of the times.
We have observed that the chancy property of a system is linked to
objective aspects of the structure of the system: the outcome of a
roulette wheel hops around, in the short term, between rouge and
noir, but in the long term we notice a rough ratio of 1:1 starts to
appear; we also notice that the structural ratio of red to black
slots in the wheel is 32:32, or 1:1. We do not judge that this is a
coincidence.
The chance of the 5 being obtained was 1/6. Our state of knowledge of
the outcome is irrelevant, since the chance is a physical property of
the system.
If, say, the initial conditions, and deterministic laws, caused the
outcome to be entirely determined, then at t = -0.1 s
the 5 was determined to occur - that event was going to happen. In
other words, in an odd-sounding phrase: The 5 was definitely going to
happen in that test, but the chance1
of getting a 5 in the test was 1/6.
The previous sentence, in ordinary English, feels contradictory. But
a chance1 does not tell us whether
a particular event is going to happen; it tells us, quite
specifically, how often this outcome occurs in a long sequence of
outcomes32. These are quite
different claims.
My superbeing example below is an attempt to clarify this point; the
superbeing, who can assess the initial conditions and do the
calculations, can know for certain that the next test will give a 5
(at the instant of release), yet also say, consistently, that the
chance1 of getting a 5 is 1/6.
We may persist: "Look, this is ridiculous. Are you saying that an
insurance company can consistently say, on the one hand: "The
chance1 of someone like you, who
has signed on for Life Insurance at age 48, dying in the next year is
0.623" (hence your premium), yet also say, on the other hand: "We
have no idea whether you will die in the next year"? Yes, we are -
this is one of Von Mises' examples. The company knows very little
about your individual circumstances - the equivalent of the initial
conditions and laws in the die case. If you ask: "Look, am I
likelyx to die in the next year?",
the company can do more than repeat what it said;
'likelyx' is just too vague - on
the one hand, of 1000 people who sign on at your age, in general,
about 623 die in the next year; on the other hand, you personally may
well have an undiagnosed fatal disease, or be very accident prone, or
whatever, and die tomorrow. Indeed, the murderer who is going to kill
you tomorrow as a result of your past actions may be completing his
final plans as you speak.
You could, in the search for a prediction as to how long you
personally are going to live, try to narrow down the collective,
alter the Unique Experimental Protocol. But either you will find that
just as it gets interesting, the data runs out, or you will find that
you end up, just as it gets interesting, with just you (in other
words, there is no longer a collective).
We may still persist: "Look, what is a reasonable degree of belief in
the claim 'I personally am going to die in the next year'? if it
isn't 0.623, what is it?" This is a clear, meaningful, question. But
unfortunately it is extraordinarily hard to answer , which is,
of course, why people are fascinated by fortune-tellers, tarot cards,
and astrology. Predicting the future life of a person, like
predicting the future behaviour of one particular particle in the
container, is beyond our present computational power. Perhaps one day
it will be possible, in which case the time-traveller, or computer,
will tell us that the reasonable degree of belief is 1; yet the
chance1 determined by the insurance
company will be unchanged33.
If the system is specified too precisely, there may be no variation
in the outcome. Doesn't ambiguity over the Unique Experimental
Protocol (UEP) - the specification of the system - make the chance
unacceptably variable, for a quality that is supposed to exist in the
external world?
Consider coin-tossing. The system which we conjecture has a chance of
0.5 of giving a head needs to be specified. A normal person (ie. one
who has not specially trained in coin tossing) is tossing a coin more
than 2 m above the floor, and projecting it upwards at least 2
m/s, with an angular velocity of 70 radians/s. This provides a UEP
for the system.
If we change the UEP, we change the system, and we change the chance.
This is an aspect of the property of chanciness.
To take the classic example: if I am considered merely as a person
aged 48, then my chance of dying in the next year, by reference to
the relevant collective, is, say, 0.7. If instead, I am considered as
a man aged 48, my chance of dying suddenly changes to 0.6. How can I
have two different chances of dying?
I am placing myself into two different systems, like a coin in a
system close to the bench (chance of heads is 0.9) and far above the
bench (chance of heads is 0.5). In the first, the entities specified
are just people. In the second, the specification has changed to men.
The social/physical/biological system which, we conjecture, generates
the collectives, differs. Since the chance is a property of the
system which generates the collective, if the system changes, and the
collective changes, then the chance changes.
There is no correct way to specify the system. I, as an individual as
opposed to 'a person', 'a man', etc, have no chance of dying next
year (as already discussed).
Suppose a deterministic macro-world. The specification of the system
- the description of what aspects of it are to be repeated (to
persist through change) - is as precise as it is. If it is very
precise, (S1), then the chance1 of
getting a 5 will be 1; if our description of S1 includes not just the
die, and the bench, but the exact position and velocity of the die as
it is released, the air currents, position of the Moon, and so on,
repeats of the test would always give 5 ex hypothesi . S1, as
specified, does not possess a chancy1 property.
If, instead, it is less precise, (S2), such that our description
merely specifies the die, and a human more than 1 m above a bench,
and leaves all other aspects of the world unspecified, then we
conjecture that repeats of the test now would give 5 1/6 of the times
(in an infinite test sequence).
This is unproblematic.
Most outcomes of S2 are different; for example, the die ends up in a
different position. What humans do is to regard "5 uppermost" as a
Natural Kind, meaning that it is of causal significance in the world,
in a way that "being 0.251 cm from the edge of the bench" is not.
Howson writes, at the end of his review article (1995 p.27): "We
clearly need a theory of objective probability, and science
positively demands one", which, considering his important work on
Bayesianism, is a weighty endorsement of the realist element in
DT .
We have presented evidence for the existence of these chancy
properties in systems, independent of human being's existence, and,
a fortiori , their degrees of belief. These properties are not
as if or in a manner of speaking , they are conjectured
to be real properties of Nature.
A way of emphasising the gulf that separates the broadly Positivist
and Realist approaches, is to consider the former's Principal
Principle : This equates the value of chance, in the
objective world, to that of a reasonable human degree of belief. If
the chance of A is r , then the
reasonable degree of belief in A is r .
Hence one might try to derive the form and the language, of chancesx
from the subjective area of degrees of belief and betting quotients -
without being committed to unacceptably low-testability chancy
properties (metaphysics); one might hope to justify regarding (Howson
(1995 p.25) "chance, as reasonable personal probability".
The Principle could thus be used by an Anti-realist as part of
a theory according to which no objective chances exist in the
external world - what exists are just unreasonable subjective
probabilities, reasonable ones, and physical events. The value of
chancesx is derived from the value of reasonable personal
probabilities.
The danger of this view is that (Howson (1995) p.20) "the lack of an
explicit argument ... for the existence of chance seems to leave the
Principal Principle with an undetermined parameter 'the chancex of
A', as the quantity which is supposed to determine our degrees of
belief"; "a proof of existence .... is lacking here".
Unsurprisingly, the Principal Principle, on our Conjecturally
Realist Dual theory, is misleading and unimportant. It merely
expresses the aim, the hope , that
mBe
- the consensus degree of belief that an event will occur - should
equal the objective chance1
Co that it will occur. We cannot
know this, because the consensus cannot get its hands on methodology
which infallibly gives 100% truth-credit to conjectures concerning
chances - in other words, which establishes that
mBc
= 1. Therefore, in most situations, with limited evidence, we may
conjecture that the true chance of an outcome is 0.5, when our degree
of belief in this outcome is considerably less.
Suppose that we have no evidence at all about a system except that it
has two possible outcomes, A and B. We have no relative frequencies,
no structural evidence, no evidence of similar systems, nothing. This
is not outcome ignorance - the short-term ignorance that is
characteristic of a chancy system. This is system ignorance -
ignorance of the nature of the system; the system could, for all we
know, not be chancy.
What is our reasonable conjectured chance for the outcome being A?
0.5? No, this is unreasonable. We have no idea what the chance
is.
What is our reasonable degree of belief, in the outcome being A? In
the absence of evidence, we do not reasonably believe either of them
to any degree. After all, if we said 0.5, we could be reasonably
asked why our degree of belief was not 0.1? What could we answer? Why
don't we believe that the outcome is always going to be B? In
this situation no choice is reasonable.
What is a reasonable betting quotient? In the total absence of
evidence, if forced to bet, we can only guess. There is no fair
bet, because we have no evidence to justify the fairness. The
idea of 'fairness', as opposed to just guessing, is that we have made
a conjecture as to the chance of the event; we have a long-term
conjecture, despite our short-term ignorance. A 'fair' bet is then
one which would ensure that, after an infinity of tests, the
better-on has not definitely gained or lost money. Conditional on the
Inductive Presupposition, it is one that would ensure that, after a
fairly long sequence of tests, the better-on has no definitely
gained or lost money. We conclude that while short-term ignorance of
the next outcome is sometimes associated with equi-chance (in the
card games and coloured ball selections of classical
probabilityx), ignorance of the
nature of the system is associated with neither equi-chance nor
equi-reasonable degree of belief.
It could be objected that we are in the odd position of having to
claim that the probabilityx of a 5
occurring on each of the two occasions of a throw of a fair die is
equal to the probability of a 5 as estimated by the long-run relative
frequency in the sequence in which those throws actually occur.
This is a successful criticism of a completely different, very
unsatisfactory, theory - a kind of Actual Frequentism, in which
chancex is a property of an event
(such as 'Coming heads'), where the value of the property is
determined by the actual frequency of the event in an observed
sequence.
More importantly, if we regard 'the long-run' as being evidence for
the relative frequency that would be obtained in an infinite
sequence, then we have a successful criticism of a Hypothetical
Frequentism, an anti-Realist version of our theory in which
chancex is a property of a
collective of events (ie. not of the system that generates it).
On our theory, there is no problem: system S1: {fair die, unaided
human thrower releasing die more than 1 m above a table} has the
chancy1 property of 1/6 of giving
5, while system S2: {loaded die, etc} has the
chancy1 property of 1/2 of giving
5. Picking up the fair die immediately changes the system, and the
chance1.
This completes my discussion of classic hurdles.
I now continue by illustrating the Dual Theory using some
defined terms, and considering how the parts of the theory vary, as
viewed by a superbeing, a clever-being, and human being.
We use Co ('o' for 'objective') to
refer to the chance1 of a 5 being
output on the next press of our device. It may be 1/6.
We use Cc to refer to the humanly
conjectured value of the chance.
We use
iBc,
('i' for 'individual'), to refer to an individual human's personal
degree of belief, her betting quotient, that a
chance1 conjecture is true.
iBc
can have any value between 0 and 1, regardless of the
available evidence. The value of her individual degree of belief in
the occurrence of a certain event, that a 5 will occur,
iBe,
follows as Cc X
Bi. For example, she may be
absolutely certain that the chance of a 5 occurring in the next press
is 1/6 (
iBc=
1), even though she has no evidence to support this. Or she may be
0.9 sure that the next press will give a 5
(iBe
= 0.9) because of the evidence of the previous sequence of numbers,
or because she is feeling lucky. iBe could be called her 'personal probability'.
Finally, we use
mBc
, ('m' for methodological), to refer to the rough degree of belief in
the truth of the conjecture (that the
chance1 of 5 occurring is 1/6),
given the evidence, prescribed by the present consensus. This is the
reasonable degree of belief. The value of the consensus degree
of belief in an event , say that a 5 will occur,
mBe
, follows as Cc X
mBc
.
The consensus is our touchstone of reasonableness. This is not
ideal, but humanity has not yet been able to think of a criterion
which is better. The 'reasonable degree of belief' is the degree of
belief that the present consensus would have, given the
evidence34.
Methodology is not yet an exact science; degrees of evidential
support, though not perhaps merely qualitative (very poor, poor, OK,
good, excellent), may well not be precisely quantitative either.
Summarising, and attempting to justify, the consensus' rough degree
of belief in a conjecture, given a certain amount of evidence - the
task of Inductive Logic - is very difficult. This, however, applies
to degrees of belief in all conjectures, not just those
concerning objective chances35.
Using this notation, we can express the following claims:
Cc = Co
is the statement of the human aim that their conjectures
should be true.
mBc
= 1 is a statement of the human hope -- not, we think, realised -
that our consensus methods of justifying chance conjectures (the ones
that define 'rational') are
foolproof36.
iBe=
mBe
is a statement of the agreement between an individual's beliefs
(judgment, and behaviour) and those sanctioned by the consensus as
'reasonable'. It states that the individual's degree of belief is
reasonable.
iBe=
mBe
= Co is a statement of the human
hope that the value of a personal probabilistic belief that an event
will occur, if rationally supported, equals the true value of the
objective chance1 of the event
occurring.
To clarify the relationship between these terms, we now consider,
in turn, a superbeing, a cleverbeing, and a human being, faced with a
world which contains chancy systems.
Superbeing : The superbeing can either perceive all past and
future events at one time, or perceive the finest details of all
physical situations - the initial conditions - and then apply the
true laws to predict its state at any later time. Either way, her
evidence is accepted by the consensus to be conclusive.
Bm, the reasonable (consensus)
degree of belief that her conjectured value for the objective
chance1, given the evidence, is true, is 1.
Cc , her conjectured value of the
objective chance1 of 5 occurring,
is 1/6.
mBc
, the reasonable degree of belief, for any being with this kind of
evidence, that Cc is true, is 1 -
as is
mBe
.
iBc
is 1.
iBe,
her personal degree of belief, her betting quotient, that the next
display will be a 5, can also be
137.
She can consistently say "the next display will definitely be a 5
(degree of belief = 1), and the
chance1 of a 5 being generated is
1/6". The appearance of contradiction is due to the confusion of
chance1 with degree of belief.
Law-but-not-initial condition cleverbeing : This is a
cleverbeing who can conclusively identify the value of the
chance1, by evidence from study of
the system, or by evidence from recording an infinite sequence
but cannot establish the initial conditions of each test
sufficiently accurately to calculate the outcome - positive feedback,
the butterfly effect, defeats him.
(i) Co is 1/6
(ii) Cc is 1/6 (his evidence
indicates this value)
(iii)
mBc
is 1; the consensus degree of belief that the conjecture that the
chance1 is 1/6 is true, given that
amount of evidence, remains 1. So the reasonable degree of belief in,
say, a 5 occurring in the next press,
mBe
, is 1/6
(v) iBc could be anything. He may have just developed an irrational obsession with
the number 5, so that he is personally certain that the next press
will give a 5.
Human being (1997 model):
(i) Co is 1/6.
(ii) Cc is 1/6 (his evidence
indicates this value)
(iii)
mBc
is less than 1. The reasonable degree of belief that this conjectured
value is the true value, depends, in some imprecise way, on the
extent of the available evidence. Despite its best efforts, the human
consensus cannot provide clear guidelines on what truth-credit to
assign to a conjecture, given a particular amount of available
evidence. In this case it might be assessed as, very roughly,
0.138.
(iv) The reasonable39 degree of
belief, betting quotient,
mBe
, that a 5 will occur next, is therefore less than 1/6. It is
Cc X Br, which is, very roughly,
1/60 - or, more sensibly, distinctly
small40.
(v)
iBe,
his personal degree of belief, could be anything. He might have
looked at the system, pressed the button 10 times, obtained: 1, 4, 5,
3, 3, 2, 1, 5, 5, 5, and decided that the next press will definitely
give him another 5, because he is gambling on it, and "he feels
lucky". In this case iBe is 1, and is unreasonable.
If instead the individual had investigated the system more
thoroughly, and consensus judged that the new amount of evidence -
extensive study of the system and extended relative frequency tests -
gave the conjecture considerable truth-credit - then the
mBc
, the methodological support, might be assessed as very roughly 0.99,
in which case the reasonable41
betting quotient
mBe
on a 5 becomes very nearly 1/6.
This is the situation that humans always face when they are making
conjectures about the world, on the basis of inadequate evidence -
whether or not the conjecture concerns chances. In the simpler case,
when the conjecture is a non-chancy1 fact or theory, the
vague, rough, reasonable degree of belief
mBc
that the conjecture is true, given the evidence, is the only
uncertainty involved, since the conjecture itself is a fact, or a
theory, involving no reference to chances.
mBe
for the event occurring, as predicted by the theory, is therefore 1 X
mBc
, since if the conjecture is true, the event will occur.
In our, more complex, case, even if
mBc
was 1 (the cleverbeing case), the consensus still would not be
able, to predict the outcome of the event with certainty, because the
hypothesis we now know to be true, only gives the event a chance
mBe
of happening .
We now return to element 2, the Epistemological part of the
dual theory. We have argued that the characteristic methods used to
obtain - I avoid 'justify' - degrees of belief in an hypothesis,
given some evidence, are not specially relevant to chance hypotheses;
these are just a special case. Nonetheless, we can now consider these
methods in a bit more detail.
WHAT ARE THE METHODS FOR GAINING TRUTH-CREDIT FOR
HUMAN CONJECTURES AS TO CHANCES
mBc?
Judging the value of
mBc,
given a conjecture, and an amount of evidence, is a methodological,
Epistemological, problem, the result of a difficulty in the human
situation.
There are two ways humans can get evidence to support conjectures of
a value for a chance1:
(a) (direct) study the situation closely, to find the characteristic
features which make it of this Kind, behave this way
(b) (indirect) collect relative frequency evidence that the
propensity is present.
The evidence will always be inadequate, for at least these three
reasons:
(i) the careful study of a system can always leave vital features
overlooked, so that our prediction of its behaviour turns out to be
completely false
(ii) an ideal test sequence needs to continue independent tests to
infinity
(iii) the experiment relies on the presumption that sceptical doubts
are quarantined; otherwise finite sequences, however long, could be
consistently misleading (ie. despite the true presence of the
chance)
How good is such evidence? How much truth-credit does it give
to the resulting conjecture? What is the
chance1that the conjecture is true,
given the evidence?
There is no decisive set of justifiedq methodological rules
establishing values for
mBc,
for chance1conjectures or for
non-chance1 ones. The situations
humans find themselves in are very varied, and not easily
summarisable by rules. No generalisable methods govern the extent of
truth credit indicated by direct evidence. Methods governing the
truth credit indicated by ratios are the procedures of Hypothesis
Testing in Statistics.
Is
mBc
subjective? It is the best assessment, by the consensus, of the
justified degree of belief in a conjecture, the truth-credit the
conjecture gains, given the amount of evidence. If it is regarded as
the chance1 that a conjecture of
that kind is true, given that amount of evidence, then it needs to be
supported itself by meta-evidence of the success of such conjectures
in the past. This pushes the problem of justifying extents of support
to the meta-level.
(a) Single values: The superbeing does not have
to justify her propositions or beliefs concerning assignment
of chances1, because she has decisive, direct, evidence; she knows
them for sure, she is always right. Humans are not in this
powerful, if slightly dull, situation. They want to make propositions
concerning the value of the chances, but they only have indecisive,
indirect, evidence. It is sufficient for them to justify/IP making a
conjecture as to the value of the chance1
Cc - with some extent
of justification
mBc
. They then can make reasonable bets; they can have some reasonable
degree of belief
mBe
that an outcome will occur.
The intellectual police cannot insist that the conjectured value
Cc, nor indeed, mBe has to equal the true value of chance1
Co. But they can insist that
people do not pretend that the evidence strongly supports/IP a
conjecture, if the consensus judges that it only very weakly
supports/IP it; they can criticise an individual if
iBe>
mBe.
Unfortunately, humanity has found that establishing consensus
guidelines for the amount of support (extent of truth-likeness,
degree of belief) justified by a given amount of evidence, is very
difficult. This is disappointing, but is not a problem for the
Descriptive Epistemologist. He simply notes it, and passes on.
(b) Several linked values : The intellectual police
can criticise people who are inconsistent in their assignments
of chances1 (for instance, such
that the conjectured chance of the six possible outcomes adds up to
more than 1). This is because such people, if committed to using
language conventionally, are asserting both something and its
contrary (In this case, both that a sequence of tests has a certain
number of outcomes, and that it has more outcomes than this).
If we started with a full description of the environment
(the values of all relevant structural parameters at that time),
which conditions (warrants) the personal degree of belief, and which
is then unchanged by further conditioning (e.g.. observed events), then
we might seem to have to end up with output
probabilitiesx of 0 or 1. The
chance seems to be 0 or 1, because if the system is fully
deterministic, and is fully specified, then it will have just one
output, the determined state. To avoid this, Objectivists may try to
include some indeterminism somewhere in the system. But this is
unnecessary.
This is a widespread error. Pierre Laplace thought that if all events
were physically necessary results of initial conditions and laws,
then nothing could be probablex in
itself - that probabilityx
depended on ignorance. Writers state that in a deterministic
world there would be no
probabilisticx propensities - that
all probabilityx is a way-station
en route to real knowledge. Yet natural determinism is
irrelevant to
chance142.
A superbeing's full description gives
iBe
= 1, but still gives Co = 1/6. Each
specific outcome is not only determined - the causal chains determine
the exact output for any specific initial state - but also
determinable by her. A human description does not obtain
mBe
= 1, because of our limitations. But both beings accept
the same description of chancy events: infinitesimal changes in the
initial conditions, at whatever time they were recorded, lead
inexorably, by deterministic laws (in a chaotic system, displaying
positive-feedback) to a certain Kind of variation in the
Poincaré variable (the Poincaré variation), which
inexorably leads, via a certain Kind of sensitive processor, to a
certain Kind of output variation. No indeterminacy exists in the
external world - yet the output shows a characteristic quality,
identifiable by a superbeing, and such that limited humans are unable
to predict specific outcomes. Each specific outcome is
determined by the initial conditions, but it is not humanly
determinable .
The world can thus be fully determinate, in the sense that each
individual outcome is determined, governed by determinate laws acting
on a system with certain initial conditions. At the same time, a
feature of the world determines that such a system, repeatedly
tested, would generate the characteristic short and long-term
behaviour. Chance1 is a successful
way of describing the outputs from this kind of system.
There is a reappearance. However, the onto-semantic analysis of
chance1 is complete, before
the epistemological analysis of degree of belief is undertaken.
Therefore it is not vicious if
chance1 reappears in this second
analysis. Thus the reasonable extent of degree of belief in a
conjecture, concerning the value of a chance in a system, could be
partly based on an assessment of the chance, given structural and
relative frequency evidence, that people get such conjectures right.
This would need to follow the same rough criteria of such
assessments. This is consistent rather than a vicious circle.
If evidence began to lead us to think that our methodological
guidelines were unsound, we would need to reconsider both basic and
meta-assessments simultaneously.
If the degrees of belief in each outcome 1-5 all
fall below 1/6, then they don't add up to 1. Does this matter?
There are two relevant degrees of belief to
consider:
(a)
mBc
This is the consensus degree of belief (betting quotient) that the
chance conjectured is true.
(b) Cc X
mBc
. This is the consensus degree of belief that a particular outcome
will be observed in the next test.
mBc
are assessments of the extent of evidence for the truth of the
conjecture that the chances1 are
1/6. They are therefore bets on the truth of the conjecture that the
chance1 of a 5 occurring is 1/6.
The betting quotient on this truth can vary reasonably from 1 right
down to 0. I can have a reasonable betting quotient of 0.1
that the true chance of a 5 occurring is 1/6 and also of 0.1
that the chance of a 4 occurring is 1/6, and so on.
Cc X
mBc
is different; it is betting not on chances but on
outcomes - whether a 5 will appear or not.
So our question is: Can we consistently accept both of the following
claims:
(i) I am sure that either outcome 1, 2, 3, 4, or 5, will appear; my
degree of belief in this composite outcome is 1.
(ii) I have very little evidence that my conjecture as to the value
of Cc is true. I could easily have wrongly assessed the system. For
all I know, the chance of 5 appearing is 0.99, or 0.11.
All that consistency requires, as summarised by the probability
calculus, is that a complete set of beliefs on the outcomes 1
to 5 should add up to 1.
Our problem was to provide an organised, summary of the human uses,
and intuitions, associated with the word 'probable' - to summarise
explicitly and truthfully the implicit ideas which are guiding the
usage. We were to suppose that people have some coherent ideas when
they use this word, but we were not to assume that a single principle
(concept) would suffice
. We were to assess our theory by its (a) consistency (b) accordance
with our uses and intuitions. We were not to presume that these human
uses, even when regarded as typically reasonable, were justified -
but instead to assess the extent of justifiability, be it high or
low.
In this Dual description of aspects of
probabilityx we have firstly
explained chance, as a Physical property of a system. Degree of human
ignorance , we have seen, is irrelevant to the description of
the system; a superbeing would note exactly the same characteristic
features of the system.
Secondly, we have described how humans obtain a reasonable degree of
belief in the truth of any conjecture, and hence, in particular, in
the chance1 of an outcome - using
rough consensus guidelines on evidential support for conjectures. We
have not tried to justify these guidelines.
We propose that this description solves many soluble extant problems
in the Philosophy of probability.
Philip Thonemann
For general references, consult pp.28-32 of Howson's excellent survey
article (Howson, C. (1995) }
References:
Howson, C. (1995) Theories Of Probability ,BJPS 46
pp.1-32
Howson, C. and Urbach, P. (1993)Scientific Reasoning: The Bayesian
Approach , Second Edition, Chicago, Open Court
Poincaré, H. (1905) The Foundations Of Science ,
Science Press, Lancaster, Pa.
Engel, E. (1992) A Road To Randomness In Physical Systems ,
(Lecture Notes in Statistics 71) Springer-Verlag
Von Mises, R. (1939) Probability, Statistics, and Truth ,
London, George Allen and Unwin
Hopf
Von Plato, J. The Method Of Arbitrary Functions , BJPS
34 pp.34-47
Gillies, D.A. (1973) An Objective Theory Of Probability ,
London, Methuen
Popper, K. (1959)The Logic Of Scientific Discovery, London,
Hutchinson
Popper, K. Conjectures and Refutations
Miller, D. (1994) Critical Rationalism Open Court
Harré, R. and Realism Rescued
Sklar, L. Physics and Chance
[Still to do: Sort out the references!]
Footnotes: (I don't
think these can be in the body of the text in HTML...)
1 I am using the subscript 'x' to mean that the word is significantly
vague, or ambiguous. Hence the subscript '1' means that I am now
using the word in a more specific sense.
2 The unitary, perhaps linguistically essentialist, conjecture 'There
is a single idea, unifying all human uses of the word
'probabilityx'
is the core of a degenerating research programme.
3 Like a Physical theorist suggesting that an observation is
mistaken, because it does not fit with her theory.
4 This is Complete Justificationism - a long-standing curse of
Philosophy.
5 Howson, in his (1995) review, reckons that the key contemporary
players are Bayesian theory of epistemic probability, and limiting
relative frequency, propensity, prequentialist, and chance, theories
of objective probability. Of these, I am not including the last two
as sub-theories. Encouragingly, he writes (p.21): "a legitimate role
for Von Mises' theory is that, combined with the Bayesian apparatus
for constructing posterior distributions, it provides the final link
between the model and reality".
6 Accepting (i) the certainty of border-region cases that it does not
cover (ii) vagueness in various of the key terms in the
description.
7 It will not, however, enable the alien to use the word
'chancex' in all everyday cases,
where its use is vague and confused, and partly determined by context
and empathy.
8 This is where we invoke a property, a propensity, to partner the
relative frequency.
9 The series can be a function of time such that the relative
frequency will not display this value. In this case, 'this machine or
this being 1+ rest of system s' produces a system S1 which is not
chancy, when 'another machine or another being 2+ rest of system s'
produces a system S2 which is chancy.
10 Without this condition, there is no identifiable system, as an
invariant, to have the various outcomes in tests (eg. the shape of
the die).
11 eg. the air currents; the velocity of the throw.
12 We can be more specific, and say that if we have 100 tests, then
the experimental ratio will lie in the range1/6 ± some small
value; if we have 1000 tests, it will lie in the range 1/6 ±
some smaller value; and so on. Indeed, we can specify how often, in
such a test run, the resulting ratio will lie outside these
ranges.
13 These commonsense descriptions were clearly expressed by Richard
Von Mises, who developed (i) the definition of the collective
(vaguely: chance as determining limiting relative frequencies) (ii)
the idea that a physical system can be conjectured to have the
property of tending to generate the collective (vaguely: chance as a
propensity) (iii) the idea that this property can be initially
defined any way we wish, because, like any other conjectured physical
property, its appropriateness will be tested by experience of Nature
(vaguely: 'chance as a theoretical entity'). We could alter the
second description so that it merely refers to "1997 human inability
to find a pattern". This would still define a perfectly respectable
property of Nature - whose full description requires reference to a
particular sensing being, just as 'whiteness' does. I am unsure if
this gives any advantage, but it is a coherent option.
14 Thus preempting the criticism that Bayesians fail to provide
justification for their helpful principled description (Miller
(1994))
15 Howson, I suggest, uncharacteristically slips up when he writes
(1995 p.16): "almost every hypothesis of use to Statistics is a
priori declared false by it {Cournot's Rule}". This would only be
true if the rule was interpreted non-methodologically, as a
restriction within the onto-semantics (the model) on the possible
consequences of the chance conjecture - as the claim that 2 5s is not
a possible consequence of C1. Such an interpretation would indeed be
incoherent interference with the model - which is why we have not
considered it.
16 Howson tells us that the weak and strong laws of large numbers are
logical consequences of Von Mises' axioms of convergence and
randomness. They will not help us in our problem in this section.
Howson, separately, hopes to prove, using Bayesian arguments, that (op.cit. p.18): "despite their infinitary character, Von Mises
collectives satisfy a criterion of empirical significance". This
enterprise, we can now see, is circular, because the presupposition
IP that underpins the crucial theorem is the very one that we are
trying to justify.
17 Or, possibly, 5 appears 360 000 times in a row - in which case the
system would appear to be of no interest at all, being like looking
at a die on a table which had 5 uppermost, recording its display,
looking away, looking back, recording its display again, and
continuing for a couple of months.
18 In other words, a + 6e produces output 1 again.
19 It may not be determinable by 1997 humans, but our extent of
evidence - our degree of ignorance - is irrelevant.
20 As we have discovered in weather forecasting, tiny variations in
the initial conditions lead to massive variations in the later state
of the system.
21 Rather than always refer to intervals, we will return to referring
only to particular values. This does not affect the argument.
22 He describes this property in terms of the analyticity of the
probability function, where 'analytic' means that the slope of the
function always exists, and varies continuously. We avoid this
formulation, but retain the concept.
23 We assume that a is the voltage associated with some value of
pressure around atmospheric.
24 The consciousness of the being, the mind, may feel that something
much freer is going on. This could be a delusion. Just as consciously
undetermined, free, actions, (random slips of the tongue) may be the
result of unconscious determined processes, so the self-consciously
random button presses, could be the result of a process in the
neurons of the brain, which does not generate some kind of real
randomness, but is just the kind of process which is being modelled
by the second system.
25 As our Physical understanding of the brain develops, we might be
able to substitute a better model here.
26 Which is interesting, but of no Philosophical significance.
27 We are assuming that she is fully aware of what every outcome of a
test will be. This does not affect the usefulness of the natural
classification 'chancy1' and the numerical measure
'chance1' to her.
28 The existence of these depends on the existence of gases, of
rivers, of people, behaving as they behave on the planet Earth. If
nothing displayed Poincaré variation, the concept of
chance1 would have no reference.
29 Thanks to Rom Harré for making this point.
30 Again we follow Von Mises.
31 ie. no gambler will be able to devise a system to beat the
odds.
32 Von Mises makes this clear. He also makes clear his fear that it
is difficult to understand - because of the hypnotic effect of
everyday language.
33 Until such information causes people's behaviour to start
changing, as it would presumably do.
34 It is open to individuals to try to change the view of the
consensus. Thus Galileo's view of the degree of belief afforded by
the evidence for the Copernican theory may have been inconsistent
with that of the consensus at that time. His task, then, was to
persuade the consensus to change. If he had failed to do so, then we
would now regard him as a crank. But he succeeded, so we regard him
as a great man. Whether we now judge him to have been 'reasonable' or
not, is a measure of our own consensus view on the weight of his
evidence.
35 We could, with some danger of regress, conjecture a value for the
chance of us getting such a conjecture right, given such amounts of
evidence. This takes us to the meta-level, at which we need to appeal
to the consensus to judge what meta-evidence we have, from the
history of investigations, as to the success-rate of such conjectures
(when they had been made with this amount of evidence) - and hence,
what truth-credit to assign to such a conjecture, what degree of
belief or confidence.
36 ie. not just 'reasonable, given the amount of evidence we have',
but 'never fail'.
37 iBe could, due to, say, an unreasonable
crisis of confidence in her abilities, be 0.1.
38 He has not got much evidence.
39 'Reasonable' in the sense that consensus meta-evidence roughly
suggests that conjectures of chances, based on that rough amount of
evidence, tend to be right about 1 time in 10. This 'reasonable' is a
matter of consensus human judgment in a situation of very limited
information.
40 These numbers for mBc are misleading. The
consensus realistically provides, at best, 'large', 'medium', and
'small', or - at least - of numbers with very large error bars.
41 A human who is judged, by a consensus, to have wildly
overestimated the truth-credit supplied by the evidence, would
produce a final subjective personal probability which would be larger
than the reasonable one.
42 Donald Gillies writes (An Objective Theory Of Probability) that
"probability theory is quite compatible with determinism". We can
explain probabilities with an underlying deterministic theory. He
rightly says (p. 136-7) that Knitchine's work, loosely following
Poincaré, only shows how macro-random processes can be an
amplification of micro-random ones. The question of what we mean by
randomness is not answered by such work; nor is the question of how
randomness originally arises.
Changes from v 2.8 (circulated June 1996)
1. Removed mistaken claim that Howson opposed a dual theory.
2. Added familiar 'long term' and 'short term' terminology (Rom
Harré)
3. Included reference to Hopf and Engel on arbitrary functions (Brian
Skyrms)
4. Restructured whole paper as Outline, 12 hurdles, various beings,
and conclusion, reducing emphasis on Physical basis for chanciness in
systems, since this is not essential to the dual theory. Removed two
examples - horse racing and die tossing.
5. Removed all comparisons with contemporary theories, due to
length.
Changes from v 5 (circulated March 1997)
1. Changed 'integrated theory' to 'dual theory' (John Welch)
2. Removed the potentially misleading terms 'objective' and
'subjective', substituting 'less subjective' and 'more subjective'
(John Welch)
3. Corrected a serious inconsistency in, and considerably clarified,
Element 2: Reasonable Degrees of Belief. The suggestion that chance
and degree of belief could co-exist in Bayes' theorem is removed. In
the process, I clarified how the Inductive Presupposition links both
the chance of a consequence to belief in a consequence, and belief in
a consequence to belief in a theory. (John Welch)
4. Changed 'probability is vague' to 'ambiguous' (Jane Hutton and
John Welch)
5. Corrected an inconsistency in hurdle 11 on Ignorance and
Equiprobabilityx. If there is no
evidence of the presence of a chance, then no reasonable degree of
belief in a particular outcome exists, and no fair betting
coefficient exists. (Again, thanks to John Welch)
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