Philosophy 5340
Topic 5: The Justification of Induction
1. Hume’s Skeptical
Challenge to Induction
In the section entitled
“Sceptical Doubts Concerning the Operations of the Understanding” in
his Enquiry Concerning Human Understanding, Hume offers, in the
following passage, an argument for the conclusion that inductive
reasoning cannot be justified:
All reasonings may be divided into two kinds, namely, demonstrative
reasoning, or that concerning relations of ideas, and moral reasoning,
or that concerning matter of fact and existence. That there are no
demonstrative arguments in the case seems evident; since it implies no
contradiction that the course of nature may change, and that an object,
seemingly like those which we have experienced, may be attended with
different or contrary effects. May I not clearly and distinctly
conceive that a body, falling from the clouds, and which, in all other
respects, resembles snow, has yet the taste of salt or feeling of fire?
Is there any more intelligible proposition than to affirm, that all the
trees will flourish in December and January, and decay in May and June?
Now whatever is intelligible, and can be distinctly conceived, implies
no contradiction, and can never be proved false by any demonstrative
argument or abstract reasoning a priori.
If we be, therefore, engaged by arguments to put trust in past
experience, and make it the standard of our future judgment, these
arguments must be probable only, or such as regard matter of fact and
real existence according to the division above mentioned. But that
there is no argument of this kind, must appear, if our explication of
that species of reasoning be admitted as solid and satisfactory. We
have said that all arguments concerning existence are founded on the
relation of cause and effect; that our knowledge of that relation is
derived entirely from experience; and that all our experimental
conclusions proceed upon the supposition that the future will be
conformable to the past. To endeavour, therefore, the proof of this
last supposition by probable arguments, or arguments regarding
existence, must be evidently going in a circle, and taking that for
granted, which is the very point in question.
(Epistemology – Contemporary Readings, ed. Michael Huemer, pages 303-4)
Hume’s argument here can be
summarized as follows:
(1) It is logically possible that the future does not resemble the past.
Therefore,
(2) There can be no deductive justification for inductive inference.
(3) Any attempt to justify induction by appealing to an inductive
inference would be circular, and would beg the question, since it would
assume that inductive inference is justified.
(4) Circular justification is not acceptable.
Therefore,
(5) There can be no inductive justification for inductive inference.
(6) The only possible ways of justifying some proposition are either by
deduction or by induction.
Therefore,
(7) There is no way of justifying inductive inference.
2. Logical Probability
and Hume’s Skeptical Challenge to Induction
One response to Hume’s argument, advanced by David
Stove in his book The Rationality of Induction (Oxford: Oxford
University Press, 1986), is that Hume overlooks the possibility of the
idea of logical probability.
What is logical probability? First of all,
logical probability is a type of probability, where probability is
something that obeys certain axioms. If we use “Pr(p) = k” to say
that the probability that proposition p is true is equal to k, then
here is one standard way of formulating axioms for probability:
Axiom 1 (Normativity):
For any p, 0 ≤ Pr(p) ≤ 1.
Axiom 2 (Necessary truths): If it is necessarily
true that p, then Pr(p) = 1.
Axiom 3 (Additivity):
If p and q are logically incompatible, then Pr(p v
q) = Pr(p) + Pr(q).
Two theorems that follow from these three axioms are these:
Theorem 1:
Pr(p) + Pr(~p)
= 1.
Theorem 2 (Overlap): Pr(p v
q) = Pr(p) + Pr(q) – Pr(p & q).
Next, let us introduce the idea of conditional
probability, using “Pr(q/p) = k” to say that the probability that
proposition q is true given only that proposition p is true is equal to
k. Conditional probability is then defined as follows:
If Pr(p) > 0, then Pr(q/p) = def. Pr(q & p)/Pr(p).
(The restriction Pr(p) > 0 is needed since division by zero is
mathematically undefined.)
Given the definition of conditional probability, one
can now prove a number of other useful theorems, including the
following:
Theorem 3 (Multiplication): Pr(q & p)= Pr(q/p) x
Pr(p)
Theorem 4 (Total Probability): If Pr(p) > 0, then
Pr(p) = Pr(p/q) x Pr(q) + Pr(p/~q) x Pr(~q)
Theorem 5 (Logical Consequence): If p => q, then Pr(p) ≤ Pr(q).
In general, then, probability is anything that
satisfies the axioms of probability. But what is logical
probability? The answer is that the concept of logical
probability is the concept of (1) a relation between a proposition and
a number that (2) is a necessary relation, rather than a contingent
one, and that (3) satisfies the axioms for the general concept of
probability.
So viewed, one can think of logical probability as
akin to the relation of entailment between propositions, with logical
probability being a more general relation. In particular, one can
think of (a) proposition p’s entailing proposition q as corresponding
to its being the case that the logical probability of q given p is
equal to one, and (a) proposition p’s entailing proposition ~q as
corresponding to its being the case that the logical probability of q
given p is equal to zero.
Whether there is such a relation of logical
probability is a controversial matter, with a number of philosophers
having argued that there is no such relation.
3. The Justification of Induction:
Two Very Different Types of Cases
As Hume posed the problem of
induction, it involved showing that it is reasonable to believe that
regularities that have held in the past will continue to hold in the
future. Since the regularities that Hume had in mind were those
associated with laws of nature, I prefer to say that the problem of
induction upon which Hume focused was concerned with the question of
how, if at all, one can prove that there are laws underlying the
regularities in question. (If one holds, as Hume did, that laws
are nothing more than certain cosmic regularities, one can rephrase
this by saying that the problem is how, if at all, one can demonstrate
that certain regularities that have held in the past are in fact cosmic
regularities, or cosmic regularities of the right sort.)
Situations are very common,
however, where one moves from information about events of a certain
sort having some property, P, to a conclusion to the effect that
further events of that sort are also likely to have property P, where
one does not think that the reason that this is the case is that there
is some underlying law. An urn contains marbles, and after
shaking up the urn, a marble is drawn which turns out to be red.
This action is repeated, say, 99 times, and in each case, the marble
drawn is red. It is then widely thought that the fact that the
first 100 marbles drawn from the urn were all red makes it more likely
than it initially was that the next marble will be red. But one
does not think that this is so because there is some underlying law
that makes it the case either that all the marbles in the urn red, or
that the probability that a marble in the urn is red is equal to k.
4. David Stove’s Approach to the
Justification of Induction
I think that induction is
justified in both types of cases. But I also think that the
justification for the inductive inference is very different in the two
cases. In the marble-and-urn sort of case, I believe that one of
the proofs of the justification of induction that David Stove offers in
his book The Rationality of Induction (1986), which is based upon the
approach of D. C. Williams in his book The Ground of Induction (1947),
and which makes use of the statistical law of large numbers, is in
principle sound. Stove advanced that argument to prove, of
course, not that inductive inferences were justified in marble-and-urn
cases, but that they were justified when what one is attempting to
arrive at are laws – or, if one prefers, relevant cosmic
regularities. In this, I think that Stove was mistaken. In
the marble-and-urn case, one is arriving at a conclusion about the next
marble drawn from the urn based on a random selection of marbles from
the urn. Any marble in the urn could have been among the marbles
drawn earlier, and I think that this is crucial. But when
one is attempting to show that it is reasonable to believe that there
are certain laws of nature, one’s observations are not being selected
randomly from the totality of events, since there is no possibility of
future events being part of one’s sample.
A way of thinking about this is
to consider a case where there are two urns containing marbles, and all
of the drawings of marbles are from the same urn. The law of
large numbers, properly formulated, can surely not be used to arrive at
any conclusions concerning marbles in the other urn. But if this
is right, then how could it be otherwise when one replaces urns with
distinct spatiotemporal regions – namely, the past and the future?
In short, arriving at statistical
conclusions about a population that one has sampled extensively, and
arriving at laws, or nomologically based probabilities, are very
different things. So, for example, if observations of drawings
involving one urn led to the conclusion that some law was involved –
imagine, for example, that the drawing of a red marble was always
followed by a drawing of a green marble, and vice versa – then one
could project that finding from the one urn to the other. But
statistical information about the colors of marbles in one urn cannot
be thus projected, and it seems to me that the same is true when
references to urns are replaced by references to distinct
spatiotemporal regions, however large the latter may be.
5. Thomas Bayes and
the Justification of Induction
The idea that the concept of logical probability is
relevant to the justification of induction goes back a long way – in
particular, it goes back to Hume’s own time. Thus, of the possible
responses to Hume’s skepticism concerning induction, the most
interesting and the most important, in my opinion, is found in Thomas
Bayes’ posthumously published “An Essay Towards Solving a Problem in
the Doctrine of Chances” (1763). But Bayes does not refer to
David Hume, who was born in 1711 and died in 1776, and I suspect that
Bayes was unaware of Hume’s argument for the conclusion that induction
is unjustified. But equally, one suspects, Hume was completely
unaware of Bayes’ argument – as are almost all present-day
philosophers, Bayesians included.
In brief, after proving a number of minor things,
including the theorem that now bears his name, Bayes went on to
consider what was once often referred to as the problem of ‘inverse
probability’, where this is the matter of determining the probable
distribution of some unobserved (or unobservable) variable given
information about the distribution of some known variable that depends
upon the unknown variable. So, for example, given information
about how many times an uneven, weighted coin has landed heads in a
given number of tosses, if one thought that the outcome of such tosses
depended upon an unobservable propensity of the coin to land heads,
determining the inverse probabilities would be a matter of determining
the probabilities of different possible propensities.
6. The Problem of
Justifying Induction
Given the idea of inference to the best explanation, briefly discussed
in an earlier lecture, it is natural to think that if one wants to find
a justification for induction, starting out from the idea of inference
to the best explanation is the way to go. It seems to me,
however, that that is not the route that one should travel, since it
seems to me that a principle of inference to the best explanation is
not at all a plausible candidate for a fundamental principle of
inductive logic. (I shall not, at this point, explain why I think
that that is so, but this is something that we can discuss later.)
6.1 Rudolf Carnap and
Inductive Logic
If it is a mistake to think of some principle of
inference to the best explanation as a fundamental principle, how
should one approach questions concerning inductive inference? Here I
have been strongly influenced by Rudolf Carnap’s book Logical
Foundations of Probability, and it seems to me that what one needs to
do is to think in terms of the concept of logical probability.
Any such system of logical probability, to be
satisfactory, has to rest upon fundamental principles of
equiprobability. Carnap, in his approach, thought in terms of two
main alternatives. One involved treating what he called “state
descriptions” as equally probable. The other involved treating
what he called “structure descriptions” as equally probable. Let
me describe a very simple type of world that will give those of you who
are not familiar with these notions an intuitive grasp of the
difference, since these two notions will be important in what follows.
Consider possible worlds where there are only three
things – a, b, and c – and only two properties P and Q, where P and Q
are incompatible with one another, and where everything must have one
property or the other. Then, for each object, there are two
possibilities: either it has property P or it has property Q. For
the three objects, then, there are the following (2 x 2 x 2) = 8
possibilities:
State description 1: a has P and b has P and c has P
State description 2: a has P and b has P and c has Q
State description 3: a has P and b has Q and c has P
State description 4: a has Q and b has P and c has P
State description 5: a has P and b has Q and c has Q
State description 6: a has Q and b has P and c has Q
State description 7: a has Q and b has Q and c has P
State description 8: a has Q and b has Q and c has Q
Each of these eight possibilities is a state description.
Next, there is the idea of a structure
description. The basic idea is that a structure description
indicates only how many things have various properties and combinations
of properties, but does not indicate which particular objects have the
various properties. So continuing with the example of worlds that
contain just the three particulars a, b, and c and the two incompatible
properties P and Q, one has the following four possible structure
descriptions:
Structure description 1: All three things have property P
Structure description 2: Two things have property P, and one has
property Q
Structure description 3: One thing has property P, and two have
property Q
Structure description 4: All three things have property Q.
Given these two ideas, one possible way of defining
logical probability is by treating all state descriptions as equally
likely, while another is to treat all structure descriptions as equally
likely. These two different choices will lead to different
results, as is clear from the fact that while structure description 1
corresponds to just the one state description – namely, state
description 1 – structure description 3 includes state descriptions 5,
6, and 7.
Of these two approaches, the first seems very
natural, while the second has no evident rationale. But Carnap
set out an argument, which we shall consider shortly, for the
conclusion that if one treated state descriptions as equally likely,
the result would be that one could never learn from experience.
As a result, he went with a definition of logical probability based on
the idea that all structure descriptions are equally
likely.
6.2 Thomas Bayes’ Essay
Thomas Bayes is, of course, a
very well known figure, with a very familiar theorem that bears his
name. But that theorem is trivial, and was only a miniscule part
of his essay. Bayes was interested in proving much more
substantial – in particular, theorems bearing upon the following
problem which he states at the very beginning of his essay:
Given the number of times in which an
unknown event has happened and failed:
Required the chance that the
probability of its happening in a single trial lies somewhere between
any two degrees of probability that can be named.
But if one can solve this problem, if one can
establish a formula relating the probability that the objective chance
of a certain sort of event lies between certain bounds, given
information about the relative frequency of events of the sort in
question, then one has done something very substantial indeed, for one
has then solved the problem of justifying induction. The
question, then, is whether Bayes solved the problem of justifying
induction, and did so in David Hume’s own lifetime, and by a method
that Hume never considered.
Now Hume, had he been aware of what Bayes had done,
might well have objected that one could not make sense of the notion of
chance with which Bayes was working. Hume would have been right
that there was no way of analyzing that notion at that time. But
it now seems clear that we are able to analyze that notion. If
so, that sort of Humean objection can no longer be sustained, and we
need to confront the question of whether Bayes did succeed in
justifying induction.
My own view is that Bayes did not quite succeed, but
that he was very much on the right track. Bayes’ basic approach
involved introducing the metaphysical idea of chances, or propensities,
and then he adopted an equiprobability principle according to which, to
put it a bit loosely, any two propensities of the same general type are
equally likely.
My objection to this type of approach grows out of
my interest in laws of nature. It is that I think that there are
good reasons for holding that objective chances cannot be ultimate
properties of things. Instead, objective chances logically
supervene on, and must be reduced to, causal laws of nature plus
categorical properties and relations. But if that is right, then
a solution to the problem of justifying induction should be set out in
terms of equiprobability principles that are formulated, not in terms
of propensities, but, instead, in terms of laws of nature.
7. Analysis and New
Alternatives in Metaphysics
The availability of a method of analyzing
theoretical terms that is compatible with a non-reductionist
interpretation of those terms opened the door not only to the
possibility, for example, of defending indirect realism as an account
of perceptual knowledge, but also to the possibility of
non-reductionist analyses of a number of very important metaphysical
notions, including the ideas of causation, of propensities and
objective chances, of dispositional properties, and of laws of
nature. Given the breakthrough in analysis, non-reductionist
analyses of all of those concepts can now be given.
The possibility of setting out such analyses does
not, of course, show that the concepts in question are metaphysically
unproblematic. Thus it could turn out that just as in the case of
the concept of a logically necessary person, where the vast majority of
philosophers think that, although an analysis of that concept can be
given, it turns out that the concept is such that it is logically
impossible for there to be anything answering to that concept, so one
might think, for example, that though one can offer non-reductionist
analyses of causation, of propensities, and of laws of nature, it turns
out, for some or all of those concepts, that it is logically impossible
for there to be anything to which those concepts, thus analyzed, truly
apply.
I shall not consider that issue here. What I want to do here,
instead, is simply to explore the relevance of this issue, in the case
of laws of nature, to the problem of justifying induction, and what I
shall argue is that is that the justification of induction stands or
falls with whether it is possible to set out a coherent
non-reductionist account of laws of nature.
8. Reductionist Versus
Non-Reductionist Accounts of Laws of Nature
One of the great divides in contemporary metaphysics
is that between philosophers who defend reductionist approaches to such
things as laws of nature and causation, and those who defend
non-reductionist approaches. So let us consider this divide, in
the case of laws of nature.
What is involved in a reductionist approach to laws
of nature? There are various ways of explaining this, but here I
think it will do simply to say that reductionist views of laws of
nature involve the acceptance of something like the following thesis of
Humean Supervenience:
All matters of fact logically supervene on states of affairs that
consist of particulars having non-dispositional properties and standing
in spatial, temporal, and spatiotemporal relations that do not involve
causation.
A reductionist approach to laws of nature, then, is
an approach that holds that laws of nature logically supervene upon
those sorts of states of affairs involving particulars. A
non-reductionist approach to laws of nature rejects this supervenience
claim.
Can a non-reductionist approach be characterized in
a more positive way, rather than simply in terms of a rejection of
Humean Supervenience? The answer is that it can be, and the sort
of account that I favor is essentially as follows:
Laws of nature are atomic states of affairs consisting of second-order
relations between properties (universals) that, first of all, are not
entailed by any set of Humean states of affairs, and that, secondly, in
the case of non-probabilistic laws of nature, entail that some specific
regularity involving Humean states of affairs obtains.
9. Non-Reductionism,
Reductionism, and the Epistemological Challenge
9.1 The Challenge to Non-Reductionist
Views of Laws of Nature
A common objection to non-reductionist approaches to
laws of nature is that, in postulating the existence of states of
affairs that involve something more than Humean states of affairs,
there is no way of justifying the belief in the existence of the extra
ontological items that are being postulated. Thus Barry Loewer,
for example, in his paper “Humean Supervenience”, claims, “The
metaphysics and epistemology of Humean laws, and more specifically,
Lewis-laws, are in much better shape than the metaphysics and
epistemology of the main anti-Humean alternatives.” Loewer
himself does not really offer much support for the epistemological part
of this claim. But this type of epistemological objection is
certainly defended by others, most notably, perhaps, by John Earman and
John T. Roberts, who devote a two-part, 56-page paper in Philosophy and
Phenomenological Research to an attempt to establish this
objection.
9.2 The Challenge Reversed
A crucial claim, then, which
reductionists with regard to laws of nature advance, is that
non-reductionist approaches to laws of nature face a serious
epistemological challenge: How can one possibly be justified in
believing in the existence of anything more that cosmic
regularities? How can one be justified in believing in the
existence of strong laws of nature, understood as atomic states of
affairs involving second-order relations between universals that are
supposed to underlie, and provide a basis for, regularities?
Reductionists with regard to laws of nature generally are confident,
moreover, that this challenge cannot be met.
In what follows, I shall attempt
to do two things. First of all, I shall refer to some things that
can be proven that together show that if strong laws of nature are not
logically possible, then a belief in reductionist laws of nature cannot
only not be justified: it can be shown to be unjustified.
Secondly, I shall then cite other results that can also be proved,
which show that if, on the contrary, strong laws of nature are
logically possible, then it can be shown that certain inductive
inferences are justified.
10. Reductionist Approaches
to Laws of Nature and Inductive Skepticism
The results that I have in mind
depend upon whether, in formulating inductive logic, one assumes that
all state descriptions are equally probable, or whether, as Carnap
thought, all structure descriptions are equally probable. I shall
not, at this point, offer a technical explanation of those two
technical notions, since I think that the mini-world example offered
earlier should suffice for present purposes.
Let me, then, simply state some
results. In doing so, I shall often refer to a concrete case of
the relevant theorem, rather than formulating it in a general and very
abstract way. Finally, all of these results are predicated on the
assumption that strong laws of nature are not logically possible.
The first two results are based
on the assumption that the correct equiprobability assumption on which
to base one’s inductive logic is that it is state descriptions that are
equally likely. Given that assumption, one has the following two
results
Result 1
Suppose, for concreteness, that
there is an urn that contains a million marbles, each of which is
either red or green. Given no information at all, what is the
probability that the millionth marble drawn from the urn is red?
The answer is . Suppose, now, that 999,999 marbles have
been drawn from the urn, and that all of them are red. What is the
probability, given that information, that the millionth marble drawn
from the urn is red? The answer is still .
Conclusion: If strong laws of
nature are logically impossible, and all state descriptions are equally
probable, then one cannot learn from experience.
Result 2
Suppose, first, that, through the total history of
the world, there are an infinite number of things that are F.
Suppose, further, that a billion things that are F have been observed,
and that all of them were G. What is the probability that that
all Fs are Gs? The answer is that that probability is either
equal to zero, or, if one accepts infinitesimals, infinitesimally close
to zero.
Suppose that, somewhat disheartened by those two theorems, a
reductionist with regard to laws of nature follows Carnap’s lead, and
defines logical probability based on the proposition that it is
structure descriptions, not state descriptions, which are equally
likely. Then one has the following theorem:
Result 3
If property G does not belong to a family of positive, incompatible
properties, then given only the information there are n Fs, and that
all of them have property G, the probability that the next F will also
have property G is equal to (n + 1)/(n + 2).
(This is Laplace’s famous Rule of Succession.)
Now this is a cheering
result. The more Fs one observes, all of which have property G,
the more likely it is that the next F has property G. So one can
learn from experience.
But one also has the following
theorem:
Result 4
Suppose, once again, that,
through the total history of the world, there are an infinite number of
things that are F. Suppose, further, that a billion things that
are F have been observed, and that all of them were G. What is
the probability that that all Fs are Gs? The answer, once again,
is that that probability is either equal to zero, or, if one accepts
infinitesimals, infinitesimally close to zero.
Results 2 and 4 look depressing if one holds that
strong laws of nature are logically impossible. But Hans
Reichenbach offered an interesting argument for the following theorem:
Result 5: If probabilistic
laws of nature are logically possible, then no evidence can ever make
it likely that a non-probabilistic law obtains.
If Reichenbach is right, then one
can never confirm any non-probabilistic law, and so Results 2 and 4
need not trouble the reductionist with regard to laws.
Happiness for the reductionist,
however, is short-lived. For, first of all, if one returns to the
idea of defining logical probabilities based on the proposition that
all state descriptions are equally likely, one can then prove the
following theorem:
Result 6
Suppose, for concreteness, 1000
Fs have been examined, and all 1000 have turned out to be Gs.
What is the probability that, if 1000 more Fs are examined, 90% of the
combined set of 2000 Fs will be Gs? The answer is that it is
2.04944 x 10-86.
This is a rather small
number. What it illustrates is that the combination of a
reductionist approach to laws of nature with a state description
approach to logical probability is not going to allow one to be able
confirm the existence of some law or other to the effect that the
probability that an F is a G is equal to k, where k falls in some
moderate interval in the vicinity of the number one.
Suppose, finally, that one shifts, once again, from
a formulation of logical probability that treats state descriptions as
equally likely to a formulation that treats structure descriptions as
equally likely. Does that save the reductionist? The answer
is that it does not, since one can prove the following theorems:
Result 7
Suppose that 1000 Fs have been
examined, and all of them have turned out to be Gs. What is the
probability that, if 1000 more Fs are examined, 90% of the combined
total of 2000 Fs will be Gs? The answer is that the probability
is just over 20%.
Result 8
Suppose, finally, that one
billion Fs have been examined, and all of them have turned out to be
Gs. Suppose, further, that in the total history of the universe,
there are an infinite number of Fs. What is the probability that
90% of all the Fs will be Gs? The answer is that the probability
is equal to 10%.
The moral, I suggest, seems
clear: if one embraces a reductionist approach to laws of nature, then
regardless of whether one adopts a state description approach to
inductive logic or a structure description approach, one will not be
able to avoid the following conclusion: No interesting scientific
hypothesis concerning laws of nature can be confirmed.
11. Families of
Properties and the Epistemology of Strong Laws of Nature
So how are things
epistemologically if strong laws of nature are logically
possible? The answer to that question depends upon the idea that
the most basic equiprobability principle is one that is formulated, not
in terms of either state descriptions or structure descriptions, but in
terms of families of properties. One way of formulating such a
principle is as follows:
Equiprobability and Families of
Properties
Given any family of incompatible
properties, if P and Q are any two members of such a family, then the a
priori probability that a has property P is equal to the a priori
probability that a has property Q.
Given such a principle, if strong laws of nature are
logically possible, and if they can take the form of certain atomic
states of affairs consisting of irreducible second-order relations
among universals, then certain sets of such second-order relations will
be families of relations. Accordingly, one can apply the
equiprobability principle just stated to such families of nomic
relations, thereby generating equiprobability conclusions concerning
laws of nature.
Given this starting point, one
can then work out the probability that a strong law of nature, falling
within a certain range, does obtain, given information about events
that would fall under such a law if it did exist. Doing this
involves a somewhat complicated calculation, especially because one has
to take into account the possibility of probabilistic laws connecting
being F with being G. But I can set out a table that will make it
evident that if one accepts the idea of strong laws of nature, the
epistemology of laws of nature is in reasonably good shape.
First of all, however, I need to
explain some notation:
“ aL1“ means “There is some number k such that it is a law that the
probability that something that has property F has property G is equal
to k, where k lies in the range from a to 1.”
“Qn “means “n particular things that have property F all have property
G.”
“M2 “means “G does not belong to a family of positive properties, so
that the only possibilities are either having property G, or not having
it.”
“Pr(q/p) = m “ means “The logical probability that q is the case given
that p is the case is equal to m.”
So “Pr(aL1/Qn & M2) = m ” means “The probability that there is some
number k such that it is a law that the probability that something that
has property F has property G is equal to k, where k lies in the range
from a to 1, given that n particular things that have property F all
have property G, and that G does not belong to a family of positive
properties is equal to m.”
Result 9
If a = 0.99, then the value of Pr(aL1/Qn & M2) is given by
the following table:
Value of n
1
0.000199
10
0.003140791
100 0.159292545
200
0.404718644
300
0.60791531
400 0.750963679
500 0.844923366
1000 0.986867122
What this table shows is that if it is possible for
there to be strong laws of nature, then the probability that it is
either a deterministic law that all Fs are Gs or else that there is a
probabilistic law to the effect that the probability that something
that is F is also G is equal to k, where k lies in the range from 0.99
to 1, can be raised to quite high values by a relatively small number
of instances.
Summing Up
A very promising approach to the
solution of the problem of justifying induction involves making use of
the idea of logical probability. But the prospects for such an
approach depend crucially upon the metaphysics of laws of nature.
The reason, as we have just seen, is that, on the one hand, there are
theorems that provide excellent reason for thinking that if strong,
governing laws of nature are not logically possible, then no laws of
nature can ever be confirmed, while, on the other hand, there are other
theorems that provide excellent reason for thinking that that if strong
laws of nature are logically possible, then the existence of such laws
can be confirmed, and thus that induction can be justified.