CHANCE
II—JOHN WATLING
A person may be more or less strongly convinced that a Channel tunnel will be
built. Presumably his degree of conviction will be related to the manner in which he
suits his actions to the rewards and penalties the various courses of action open to him
will bring if his judgment about the building of a tunnel proves correct, and the
rewards and penalties those same actions will bring if his judgement proves incorrect.
However, I do not think that his degree of conviction can be identified with this
manner since a person will act in face of a higher penalty in a rash mood than in a
prudent one. Nor do I think that this manner of suiting his actions to the various
rewards and penalties they will, with different eventualities, bring can be measured by
discovering the least favourable odds which will induce him to place a bet. The least
favourable odds which will induce him to place a bet vary with the size of the bet and
with the impression the prospective gains and losses would make upon his financial
situation. We test the seriousness of a person’s assertion by asking how much he will
bet, not what odds he will take.
There seems no reason why one person should not recommend others to take
up a certain degree of conviction in a certain matter. Perhaps if one person said to
some others “It is certain that a Channel tunnel will be built” he would he
recommending them to adopt a very high degree of conviction in the building of the
tunnel. Perhaps if he said “It is just possible that a Channel tunnel will not be built” he
would be recommending them to moderate their strong conviction that a tunnel will
he built by a small degree. Mellor identifies the question whether there is an objective
probability of a tunnel being built with the question whether there are sufficient
grounds for recommending one coherent betting quotient rather than another.
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Further, he identifies the objective probability of a tunnel being built with the fact
which constitute those grounds. This fact, he assumes, concerns a particular feature of
the world. It can hardly be that the objective probability of a tunnel being built is
identical with a fact which includes the financial position of the people to whom a
recommendation is addressed, nor that a recommendation concerning how prudent
to be - as opposed to one concerning how to be prudent - depend upon any fact other
than one concerning a non-natural property. Therefore, if objective probability is to
be approached in this way at all, the question whether there is an objective probability
that a tunnel will be built must be identified with the question whether there are
sufficient grounds for recommending one degree of conviction rather than another,
and the objective probability itself must be identified with the fact which constitutes
the grounds for recommending that degree of conviction. How plausible are these
identifications?
I do not think that the second is plausible at all and without the second the first does
not help very much to explain what objective probabilities are. The fact that nearly
everyone in the world wanted a tunnel, that resources were available and engineering
skill adequate, would be a sufficient ground for recommending a high degree of
conviction that a tunnel would be built, yet it would not he the fact that it was very
probable that a tunnel would be built. It would not be that fact since it is also true that
the fact, if it were one, that some very powerful financiers would gain if the tunnel
were built and no very powerful financiers would gain it the tunnel were not built
would be a sufficient ground for recommending a high degree of conviction that a
tunnel would be built, so that that fact, if it were one, would be the fact that it was very
probable that a tunnel would be built. The same fact could not, even in different
circumstances, be identical with each of two different facts, hence Mellor’s second
identification is mistaken. Even it were not, what ground would there be for
supposing that all facts which provide sufficient grounds for recommending a degree
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of conviction of a certain strength have it in common that they concern one particular
feature of the world?
I do not think that the first identification is plausible as it stands; perhaps it
could be made more plausible by specifying the relation of being a sufficient ground
precisely. If nothing is a sufficient ground for recommending a conviction of a certain
strength which does not entail that a conviction of that strength ought to he
recommended then—in my opinion—Mellor’s first identification has the consequence
that there are no objective probabilities. If, as seems more likely, the question of
whether there is a sufficient ground for recommending a conviction of a certain
strength rests on considerations of morality, then the first identification makes the
existence of objective probabilities entail the existence of moral principles of a certain
sort. I find that implausible. I shall not pursue this matter since I do not think that
Mellor treats either identification very seriously. If he did he would not assume
without argument that all facts which provide sufficient grounds for recommending a
degree of conviction of a certain strength have it in common that they concern one
particular feature of the world, nor that that particular feature is the subject matter of
the statistical sciences, by which he means those sciences which are concerned with
hypotheses of statistical, rather than universal, form. He does take two things to be
important: first, that the fact that people sometimes have coherent betting quotients
does not imply that there are no objective probabilities; second, that problems arise
over such propositional forms as this—which he takes to be typical of propositions
implied by hypotheses of statistical form—that the chance that this F is a G is 1/n
where n is greater than one). I agree with him about both. He uses the term “chance”,
rather than the term “probability”, to make clear that the probabilities in which he is
interested belong in no sense to logic.
His first characterization of the propositions concerning chance he wishes to
investigate is that each ascribes the chance of a certain outcome to a trial. A trial is a
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particular performance of an experiment. Mellor does not make clear whether chances
are ascribed to the event of an experiment being performed or to the fact that an
experiment was performed, although one of his examples suggests that he intends the
latter. The distinction is important, as Peter Downing has pointed out in discussing
causation. ‘‘His arrival at two o’clock caused a sensation” invites the question “What
was it about his arrival at two o’clock which caused the sensation? Was it the fact that
his arrival took place at two o’clock, or merely the fact that it took place at all, or was it
some other fact about it?” Events are particulars; strictly, they can no more be causes
than physical objects can, but an event or a physical object may be said to be
responsible for an effect in order to suggest that some fact about it had that effect, or
that there is some fact about it which had that effect. This is the foundation of jokes of
the type: “What gave you that black eye?” “My wife hitting me with a pound of
tomatoes.” “But your wife hitting you with a pound of tomatoes wouldn’t give you
that black eye.” “They were in a tin.” Mellor does not wish to confine his attention to
contrived experiments, rather than natural happenings, so that this characterisation
does little more than limit the ascription of chances to facts to the effect that
something has happened at a time or for a period of time. However, Mellor sometimes
seems to be misled by his use of the term “trial”. When he is discussing the probability
that a radium atom will disintegrate he speaks of the trial as that of waiting for a year.
What chance there is that the fact that I wait for a year will have the outcome that an
atom of radium disintegrates depends, of course, upon what chance there is that the
fact that I wait for a year has the outcome that a radium atom exists at the beginning
of that year. However, the chance that a radium atom will disintegrate does not
depend upon that. The trial Mellor has in mind must be the fact that I wait for at most
a year with a radium atom. However, if this is the trial, the fact that I wait is redundant
to it. The chance Mellor wishes to express is the chance of the fact that a radium atom
exists at a certain time having as outcome its disintegration within the year. I suspect
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that he avoided this formulation because he wishes to regard trials as experiments on,
or happenings to, objects. The continuing in existence of an object is neither an
experiment on it nor something which happens to it.
It is not clear how much of their ordinary meanings the terms “outcome” and
“result”—an outcome for Mellor is a disjunction of results—are intended to preserve.
It may be that if it is true that since an experiment has been made at least one, and no
more than one, of a group of events must occur and not true that since the experiment
has been made some particular one of those events must occur, then those events are
called results of the trial. However, it may be that a trial is the performance of an
experiment and if whenever that experiment is performed an event belonging to one,
but to no more than one, of a group of kinds of events occur, and if it is not true that
whenever the experiment is performed an event belonging to some one of that group
of kinds invariably occurs, then the occurrence of an event belonging to one of those
kinds is said to be a result of the experiment. Therefore it may be that the terms
“outcome” and “trial” are so used that to call an event the outcome or the result of a
trial is not to imply that its occurrence was in anyway due to the trial. To call an event
the outcome or result of a trial may not even imply that it follows the trial in time.
Disintegrating during a year does not follow waiting for a year, although, of course, it
does follow beginning to wait for a year. These questions are not of great importance
in the context of Mellor’s paper. What is important is his declaration that:
the chance of its happening is not a property that can be ascribed to an event
without taking it to be the outcome of a trial. (Section 3)
For example, we cannot say “There is a chance of 1/n that a Channel tunnel will be
built in this century” but only “There is a chance of 1/n that the existence of the
English Channel throughout the remainder of this century will have as outcome that a
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tunnel will be built under it”, or any number of other things of that sort. We must
always speak of the chance that a trial will have an outcome, or of the chance of an
outcome with respect to the occurrence of a trial. Evidently this allows that the
occurrence of an event of a certain kind may have one chance when it is considered as
the outcome of one trial and another chance when considered as the outcome of
another. This compatibility accords with the logic of propositions concerning chances.
The fact that there is a chance of 1/6 that the landing of a die will have as outcome the
turning up of a five is consistent with the fact that there is a chance of 1/3 that the
landing of the die with an odd numbered face upwards will have as outcome the
turning up of a five. Again, the fact that there is a chance of 1/2 that the existence of a
radium atom at a certain time will have as outcome the disintegration of the atom
within a period of 1,622 years is consistent with the fact that there is a much greater
chance that the existence of a radium atom under bombardment by a stream of alphaparticles at a certain time will have as outcome the disintegration of the atom within
that same period. Of course, the compatibility of these different chances is not implied
by Mellor’s contention that the chance of an event’s happening is not a property
which can be ascribed to it without taking it to be the outcome of a trial; it is merely
that unless Mellor’s contention is correct it is difficult to see how these chances could
be compatible. In any case Mellor argues at the end of his paper that the fact that the
occurrence of an event of a certain kind has a certain chance with respect to one trial
is not inconsistent with the fact that such an event is the invariable concomitant of
another kind of trial, so it seems that he accepts the compatibility. Neither of these
contentions is consistent with his belief that the frequency theory is refuted by the
argument Ayer brings against it. Ayer asks for the probability of his living to the age of
eighty and complains that the frequency theory tells him the probabilities of his living
to that age with respect to his being an organism, a human being, a professional
philosopher etc., but never the probability of his living to the age of eighty. Mellor
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provides a way of regarding this probability as the chance of a trial having a certain
outcome when he discusses the chance of a radium atom disintegrating and the
concept of an individual’s physiological age. He describes the trial as “waiting for a
certain period of years”. I have argued that it is in fact the existence of Ayer at the
beginning of a period of years. The frequency theory has certain difficulties in
interpreting the chance that this trial will have the outcome that Ayer is alive at the
end of the period, hut none of them is the difficulty to which Ayer points. He would
be just as dissatisfied with the probability of his survival with respect to this particular
trial as he is with probabilities with respect to other trials. The frequency theory has
difficulty in interpreting the probability that Ayer’s being alive at the beginning of a
period of years will have the outcome that he is alive at the end of it because although
it is possible to find a denumerably infinite sequence of such periods of years at the
beginning of which Ayer is alive, yet the two outcomes—alive at the end and not alive
at the end—are not randomly distributed. The earlier beginnings have the outcome
alive at the end, but after a certain point all beginnings have the outcome not alive at
the end. If this is Mellor’s objection to the frequency theory then it is quite consistent
with his contention that in one respect the frequency theory is correct: it regards
chance as a property that cannot be ascribed to an event without it taking to be the
outcome of a trial. However, Mellor’s argument in fact follows Ayer’s closely. What is
more he agrees with Ayer that the objection to the frequency theory is the analogue of
an objection to the theory that probability is a logical relation between evidence and
conclusion. The only objection to that theory which could be an analogue of an
objection to the frequency theory is the objection that it makes it impossible to speak
of the probability of a conclusion, and only allows that a conclusion may be probable
with respect to certain evidence. This rejection of the logical relation theory is,
therefore, based on an assumption which Mellor explicitly denies.
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This inconsistency in Mellor’s paper seems to me to be mitigated by another
which is more central to the theory which it is the main object of his paper to propose.
Once he has characterised the propositions in which he is interested as those which
ascribe the chance of a certain outcome to a trial he goes on to investigate the place of
the concept of a disposition—that is, of an object having a certain disposition—in the
concept of chance. To this end he considers the possibility that there being a chance of
1/n of a trial having a certain outcome might be regarded as the possession of a
dispositional property by the object, or device, or arrangement of objects, upon which
the trial is conducted. His question is: What is this dispositional property? One
answer, the discussion of which Mellor had to delete from his paper from lack of
space, is that there being a chance of 1/n of a trial having a certain outcome can be
regarded as a property of the object on which the trial is conducted, but not as a
dispositional property of that object. The property is closely related to a dispositional
property, but it is a tendency, not a disposition. For example, the proposition that
there is a chance of 1/6 that the falling of a die will have the outcome of a five turning
up will be interpreted as the proposition that the die tends, with a strength of 1/6, to
turn up five when it falls. Of such a proposition Mellor said:
And it is not at all clear that any analysis of it can be given that does not involve
essential reference to chance or, as above, reduce essentially to relative
frequency.
It is indeed not clear how tendencies can be interpreted except in terms of frequencies
or in terms of such relations as that of tending to produce or promoting. I am thinking
of example such as “Weight low down promotes stability”. It is the analysis of such
propositions as this which seems to me to be the most interesting question concerning
probability. Hume’s attack on the idea of causal necessity, that is, on such concepts as
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producing or bringing about, has convinced very many philosophers that these causal
relations hold between two kinds and not between two facts. If causal relations hold
between kinds they can be given an extensional interpretation. For example, a typical
causal form would be that being F causes being G and this can be identified with the
form that there never has been and never will be anything which is F and is not
G. This universal proposition has as consequences numerous relations between two
facts—for example, that this is either not F or is G—and it is easy to suppose that it
has as consequences numerous causal relations between two facts—for example, that
the fact that this was F caused it to be G. However, propositions of this form, since
they are not of the form “being F causes being G”, have no interpretation. Causation,
according to this extensionalist account, is not a relation between two facts. The words
“His being F would cause him to be G” might be taken to express the proposition that
there never has been and never will be anything which is F and is not G, but to do that
is to pretend that causation is a relation between two facts, not to make it one. I am
convinced that there are causal relations which stand between two facts and so I reject
the extensional interpretation. I find the fact that causal relations are necessary
relations to be an independent, and sufficient, ground for rejecting it.
Both of these objections tell equally strongly against an extensionalist
interpretation of the relation of tending to produce or promoting. A frequency
relation between two kinds of event cannot stand between two facts. A frequency
relation is an extensional relation which partakes in no way of necessity. I am not sure,
when Mellor asks
But if frequentists deny that their definition applies to the single case, what does
it apply to? What sense can be made of “the chance that an F is a G” that denies
sense to “the chance that this F is a G”?
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whether he has the former of these objections: in mind. It is a telling question but
slightly unfair. The frequency theory defines “the chance of being G with respect to
being F”. It defines neither “the chance that anything is G with respect to its being F”
not “the chance that this is G with respect to its being F”. This is analagous to the
extensional interpretation of causation which defines “being F causes being G” but
defines neither “this is F causes this to be G” nor “anything’s being F causes that thing
to be G”. For some reason the fact that frequencies cannot relate two facts is much
more obvious than the fact that regularities cannot relate two facts and the objection
that the extensional account of causation cannot allow causation between two facts is
much less common than the objection that the frequency theory cannot allow a
probability relation between two facts. Perhaps it is because according to the regularity
theory of causation causal propositions do have as consequences relations between
two facts, although not causal relations, while according to the frequency theory
probability propositions have as consequences no relations between two fact
Somehow this latter objection gets mixed up with the objection that the frequency
theory is a relational theory. This happens, I think, in Ayer’s discussion of the
frequency theory.
However that may be, Mellor rejects those versions of the dispositional theory
of probability which are in reality, not dispositional theories, but tendency theories.
Because he rejects any intrusion of frequencies into the concept of chance he equally
rejects, but without discussion, theories such as Peirce’s and Popper’s which do regard
chances as dispositions of objects, or of arrangements of objects, but as dispositions to
behave in a certain way not in a single trial but in a very large number of trials. Peirce
says in the paper Mellor cites
10
Now in order that the full effect of the die’s “would be” may find expression, it is
necessary that the die should undergo an endless series of throws from the dice
box.
Again, Popper writes of probabilities, in “The Propensity Interpretation of
Probability”:
They characterise the disposition, or the propensity, of the experimental
arrangement to give rise to certain characteristic frequencies when the
experiment is often repeated.
It is not clear that these formulations succeed in presenting such facts as that a die
turns up five when it fails with a frequency of 1/6 as facts about a disposition of the
die. Steel has the dispositional property of becoming hard on repeated blows with a
hammer, but there the repeated blows produce a single effect. Mellor’s example of the
elasticity of a spring is an example of a dispositional property where a single
dispositional property is allowed to be revealed in different responses to different
forces. Perhaps, then, if it is a fact that a die responds to 1200 throws with roughly 200
turnings-up of five, and to 2400 throws with roughly 400 fives, and so on, we might
regard it as having a dispositional property. What we cannot do is regard the
probability as a property of a single experiment as Popper suggests when he says that
the main point of a change to the propensity interpretation of probability is:
that we now take as fundamental the probability of the result of a single
experiment.
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Perhaps Popper was led into this inconsistency by the solution he had in mind for
problems concerning probabilities in the quantum theory. However, he often returns
to his former manner of speaking of probabilities as dispositional properties of
objects, or arrangements of objects. It is unclear to me what advantage this
dispositional manner of speaking has, except perhaps that, as Peirce’s phrasing
reveals, it allows subjunctive conditionals—and hence perhaps an element of necessity
—to be introduced into a frequency analysis.
Having rejected both of these accounts Mellor puts forward his own. It makes use of
neither tendencies nor frequencies. He holds that if an object has a disposition to
become G in circumstances F, then if ever it gets into circumstances F it will become
G. Therefore he argues that if there is a chance of 1/6 of the falling of a die having as
outcome the turning up of a five then the die is not disposed to turn up five when it
falls: it is disposed to take up a chance of 1/6 that it will turn up, or perhaps has turned
up, five. The analogue of the breaking of a glass when dropped is the establishment of
a certain chance of turning up five of a die when dropped. That is—although Mellor
never puts it like this because he has appropriated the word “result” to mean “noninvariable result”—just as the result of dropping a glass is its breaking, so the result of
dropping a die is the establishment of a certain chance of five turning up. I cannot
resist the comment that it is rather disingenuous of Mellor to claim that by this
account the chance of a trial having a certain result is analysed
in terms of a feature of a more familiar kind, namely a dispositional property of
a persisting physical entity (Sect, 4).
It is disingenuous because the disposition is a disposition to take up a certain chance
of that result, and that latter chance remains quite unexplained. However, there is a
more serious fault in Mellor’s view than that. Since Mellor analogizes the coming into
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being of a certain chance of a result with the breaking of a glass he cannot maintain
his view that a chance of 1/n of turning up five is “not a property that can be ascribed
to an event without taking it to be the outcome of a trial”. Breaking is an event which
can be ascribed to a glass without ascribing it as a result of dropping, or knocking, and
so a chance of 1/n of turning up five, since it is now regarded as a state which a die
comes to be in when it is dropped, can be ascribed to a die without mentioning
dropping, or any trial. In short, Mellor analyses “There is a chance of 1/6 that the trial
of dropping this die will have the result of turning up five” as “This die is so disposed
that when it is dropped it takes up a chance of 1/6 of turning up five”. It is valid to
argue from this chance together with the additional premise that this die has been
dropped to the conclusion that it has taken up the chance of 1/6 of turning up five.
The conclusion of this argument is an ascription of a chance to an event which does
not take it to be the outcome of any trial.
This inconsistency is a serious matter, for it is the relative concept of chance
with which Mellor began that is involved in the chances which are expressed by the
statistical hypotheses in which he is primarily interested. It is relative chances with
which the probability calculi enable us to reason, and Mellor employs the classical
calculus in his arguments in Section 2. It is true that it is non-relative chances which
would provide an objective state of affairs on which to base a bet. Mellor argues the
opposite but he is mistaken.
When a person bets that horse A will win a race he enters into an agreement. It
might for example be one according to which if the race is run and A wins he will
receive a sum of money, if the race is run and A loses he will pay out a sum of money,
and if the race is not run no financial transatlantic will take place. What he has to
decide in order to place his bet is the conditional “If the race were run, would A win?“
Now the probability that if the race were run A would win is not the same as the
probability of A’s winning on the race being run. If it were, then the fact that the
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probability of A’s winning on the race being run differed from the probability of A’s
winning on the race being run in the rain would imply that the race was not run in the
rain. The whole point of the “on” or “with respect to” phrasing is to express
probabilities that do not allow detachment.
There seem to be other strong objections to Mellor’s dispositional analysis of
the chance of a trial’s having a certain result. If chances can be properties of objects
which arise from doing certain things to them, it must be possible that they should be
causally related to other properties of objects. In particular, since they do not logically
imply the possession of the property whose chance they are, they might be causally
related to the possession of that property. Yet could it be that the chance of 1/6 which
this die has of turning up five could cause the die to turn up five? This must be a
logical possibility on Mellor’s view, but it scarcely seems to be one.
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