Justifications of induction, and certainly a priori justifications\ud
of induction, are out of fashion these days. In a\ud
chapter of his recent book, In Defense of Pure Reason\ud
(1998)1, however, Lawrence Bonjour, the respected American\ud
epistemologist, bucks the trend and makes a valiant\ud
attempt to revive the latter. What he claims can be justified\ud
a priori is that if the premise of a standard inductive argument\ud
obtains, then it is likely or probable that the conclusion\ud
will hold. A standard inductive premise, for Bonjour,\ud
will state that a certain proportion m/n of observed cases of\ud
A have been cases of B, as well as specify that there has\ud
been "suitable variation of the collateral circumstancesâ€?\ud
and that the "observed proportion ... converges over time\ud
to the fraction m/nâ€? (Bonjour, 206-07). The standard\ud
inductive conclusion will state that there is "a corresponding\ud
objective regularity in the worldâ€? (212), in other\ud
words an objective regularity of the form: m/n of all As are\ud
Bs

Fuller, Gary

Meixner, John

English

Elektronisch archivierte Theorie - Sammelpunkt

  227
Bonjour’s A Priori Justification of Induction 
John Meixner and Gary Fuller, Mount Pleasant / Michigan 
Justifications of induction, and certainly a priori justifica-
tions of induction, are out of fashion these days. In a 
chapter of his recent book, In Defense of Pure Reason 
(1998)1, however, Lawrence Bonjour, the respected Ameri-
can epistemologist, bucks the trend and makes a valiant 
attempt to revive the latter. What he claims can be justified 
a priori is that if the premise of a standard inductive argu-
ment obtains, then it is likely or probable that the con-
clusion will hold. A standard inductive premise, for Bonjour, 
will state that a certain proportion m/n of observed cases of 
A have been cases of B, as well as specify that there has 
been “suitable variation of the collateral circumstances” 
and that the “observed proportion ... converges over time 
to the fraction m/n” (Bonjour, 206-07). The standard 
inductive conclusion will state that there is “a corres-
ponding objective regularity in the world” (212), in other 
words an objective regularity of the form: m/n of all As are 
Bs. 
Since Hume’s day a priori justifications of induction have 
been received with scepticism, but the alternatives are not 
altogether satisfying to many of us, to say the least.2 It 
would be exciting if Bonjour’s project were successful. 
Sadly, as we shall show, his project fails; indeed, once the 
hidden assumptions are brought to light, we shall see that 
it seems to fail for all too familiar reasons.  
Part 1 of our paper will sketch Bonjour’s overall strategy, 
a two-step procedure, and argue that the first step, the 
“some-explanation” step, is crucial. Parts 2 and 3 will 
criticize this crucial step: Bonjour’s supposedly a priori 
inference to some explanation commits the fallacy of false 
dilemma and Bonjour shows us no way of excluding the 
intermediate alternatives (2); nor will couching his 
argument in terms of possible worlds help to make it more 
persuasive (3).  
We need to make two quick preliminary points before we 
begin. First, for the sake of the discussion we shall go 
along with Bonjour and assume that there is a workable 
notion of a priori justification, which is reasonably close to 
the traditional one of having a reason that does not 
depend on experience. Second, although Bonjour’s 
standard inductive premise and conclusion are put in terms 
of the general case of a fraction m/n of As being Bs, for the 
sake of clarity and simplicity we shall concentrate on the 
case where m/n = 1, namely the case of all As being Bs. 
From now on, then, when we refer to the standard 
inductive premise we shall be focusing on the case in 
which all observed cases of A have been (observed) cases 
of B in a wide variety of circumstances (and hence the 
observed correlation apparently converges to the fraction, 
m/n = 1); and when we refer to the standard inductive 
conclusion we shall be focusing on the claim that there is 
an objective regularity of the form: all As are Bs. 
                                                     
1 Bonjour gives a simplified and abbreviated version of this chapter in his latest 
and more introductory book, Bonjour 2002. 
2 We are thinking of such alternatives as ordinary-language, pragmatic, 
reflective-equilibrium, and reliabilist justificatory attempts, associated with such 
philosophers as Peter Strawson, Reichenbach, Goodman, and Alvin Goldman. 
1. Bonjour’s Overall Argument 
Suppose that without exception copper has been observed 
to melt at 1083 degrees centigrade in a wide variety of 
circumstances (which include different times, places, and 
experimenters). Here we have what Bonjour calls a stan-
dard inductive premise: all observed As (copper objects) 
have been observed to be Bs (to melt at 1083 degrees 
centigrade). Bonjour’s project is to show that from such a 
premise one can derive a priori a standard inductive 
conclusion, namely the “straight” conclusion that it is an 
objective regularity (a law-like regularity) that all As are Bs, 
in this case that it is a law that all copper melts at 1083 
degrees centigrade. 
Bonjour proposes to move from the standard inductive 
premise to the standard inductive conclusion in two steps, 
both of which need of course to be a priori. The first step 
argues that there must be some explanation other than 
chance to account for the inductive premise, since the 
chance hypothesis would be so improbable as to be 
miraculous. The explanation in question will cite some 
objective regularity, but not necessarily the straight law, As 
cause Bs. Instead the relevant law might be of the form 
“Being A + C causes being B” or of the “fork” form “Cs 
cause both As and Bs” (206-09). The second step argues 
that the conjunction of the inductive premise with the claim 
that there is some explanation makes it most likely that the 
straight explanation obtains (209-13).  
Bonjour’s two step procedure is reminiscent of the idea 
suggested by Hume and then further elaborated by 
philosophers such as Mill and, more recently Mackie 
(1979: 123), that one might first try to establish something 
like a principle of the uniformity of nature and second, 
uniformity having been secured, go on to infer from the 
empirical data what particular laws there are.3 Of course, if 
there is any consensus on Hume’s views it is that Hume 
was sceptical about the possibility of demonstrating a priori 
any uniformity principle. 
Bonjour’s first step of establishing his claim that there is 
some explanation for the inductive premise is really the 
crucial one. We shall be concentrating on it. If we could 
establish that there is some explanation for the observed 
regularity we would be well on our way to establishing the 
straight explanation. But how can we establish, indeed 
establish a priori, that there is some explanation? It is 
important here to see that Bonjour is not trying to establish 
a priori the categorical claim that for any event that 
happens there is an objective law that covers it. Rather 
what he wants to establish is the conditional claim, “In a 
situation in which a standard inductive premise obtains, it 
is highly likely that there is some explanation (other than 
mere coincidence or chance) for the convergence and 
constancy of the observed proportion ….” (208). Let us call 
this the some-explanation principle. 
Can Bonjour’s some-explanation principle be justified a 
priori? The rest of our paper will argue that it cannot, or at 
least that Bonjour has failed to show that it can.  
                                                     
3 Indeed, if the uniformity principle were restricted enough, it might even be 
possible to deduce from such a principle, in conjunction with empirical evi-
dence, that certain particular causal laws hold! 
Bonjour’s A Priori Justification of Induction - John Meixner and Gary Fuller 
 
 
 228 
2. Can the Some-Explanation Principle be 
Justified A Priori? 
Bonjour’s some-explanation principle, which he thinks can 
be justified a priori, is that if a standard inductive premise 
holds then it is highly likely that there must be some 
explanation other than chance for the observed regularity.  
How does he think that this principle can be justified a 
priori? His idea is that, helping ourselves to the familiar 
tools of the probability calculus, including Bayes’s Theo-
rem, and assuming that these are a priori (under approp-
riate interpretations), we can reason as follows. Suppose 
that a standard inductive premise holds, say that all 
observed As have been observed to be Bs in a wide 
variety of circumstances. What hypothesis best accounts 
for this regularity? There are two possibilities. Either the 
observed correlation is due to chance or there is some 
explanation, some objective regularity, which accounts for 
it. “Of course, [says Bonjour] it is logically possible that the 
results in question represent the operation of nothing more 
than mere random coincidence or chance, but it seems 
evident, and as far as I can see evident on a purely a priori 
basis, that it is highly unlikely that only coincidence is at 
work … ” (208).  
Here is a simple analogy. I flip a coin 1000 times and it 
turns up heads every time. Either the coin is a fair, or in 
other words chance is operating, or the coin is biased 
(some explanation is at work). It is obvious that it is 
extremely unlikely that the chance hypothesis holds here. 
This will be true even if the initial, or prior, probability of the 
chance hypothesis is high (suppose that before I flipped 
the coin I had strong evidence that it was freshly govern-
ment-minted). Since the only other possibility is that the 
coin is biased, the hypothesis that this is so (and thus that 
there is some explanation) is highly probable. 
This reasoning looks good, does it not? Unfortunately, 
there are flaws, some of which may be irreparable. At the 
very least Bonjour gives us little guidance in how to repair 
them. 
The main flaw in Bonjour’s reasoning above is that it 
commits the fallacy of false dilemma. The some-explana-
tion hypothesis, interpreted in the needed way, and the 
chance hypothesis are not exhaustive. There are many, 
indeed an infinite number, of intermediate hypotheses. 
Types of intermediate hypotheses, most quite familiar, 
include: Goodman’s grue-like hypotheses, wild curve-fitting 
hypotheses (Swinburne, 2001: 83), and hypotheses that 
invoke temporally or spatially limited objective regularities, 
e.g. the hypothesis that there is a causal law, As cause Bs, 
which holds only for a million years starting with 10,000 BC 
(Mackie, 1979: 125). These intermediate hypotheses are 
of course more or less “crazy” in the sense that they would 
be not be entertained, or if entertained then quickly 
dismissed at the outset, by common sense or science. 
Bonjour, however, needs not only to dismiss all such 
intermediate hypotheses, but also to dismiss them a priori, 
if his reasoning is to go through. 
But why does Bonjour’s some-explanation hypothesis 
have to be restricted to “sane,” or “normal,” explanations? 
Briefly, this is because without such a restriction Bonjour’s 
second step to the “straight” conclusion, namely, that it is 
an objective regularity that all As are Bs, will not succeed. 
Suppose that all examined emeralds have been observed 
to be green in a wide variety of circumstances. Such 
evidence may well make the “sane” straight explanation --- 
it is a law that all emeralds are green --- much more 
probable than the “sane” fork explanation --- there was 
some third factor present in each case that caused the 
object to be an emerald and also caused it to be green. 
That evidence, however, will do nothing to eliminate the 
“crazy” explanation that it is a law that all emeralds are 
grue. 
Bonjour, then, is going to have to eliminate a priori these 
crazy hypotheses. Can he do so? This is really the old 
question, for which the probability calculus and Bayes’s 
theorem in themselves give us no help, of how we should 
assign the initial, or prior, probabilities to possible hypo-
theses. Traditional suggestions for assigning prior proba-
bilities include appeals to background common-sense or 
scientific beliefs, to considerations of simplicity, to the prin-
ciple of indifference, or to various combinations of these. 
There has been and still is much controversy over how to 
formulate clearly such suggestions, e.g. how to formulate a 
criterion of simplicity which squares with our intuitions. 
Further, there is additional scepticism over the status of 
the connection between the suggested conditions and 
probability. It may well be that simpler hypotheses are 
more likely to be true, but is this a priori?4  
A careful examination of Bonjour’s attempt at an a priori 
justification of induction leads us back to these old 
controversial questions. Well and good. But at this point 
Bonjour leaves us hanging. The most he gives us in the 
way of an a priori rejection of crazy hypotheses are a 
footnote dismissal of “grue”-like predicates as having “any 
major bearing on the classical problem of induction” (189) 
and a short dismissal of hypotheses that postulate a 
change in the laws of nature (such as our million-year 
hypothesis above) by claiming, but not defending, the view 
that objective regularities are in some sense non-Humean 
(214-15).  
3. Do Possible Worlds Help? 
Bonjour at one point tries to couch his argument for the 
some-explanation principle in terms of possible worlds. 
“The relevant claim,” he says, “would be that it is true in all 
possible worlds that there is likely to be a non-chance 
explanation for the truth of a standard inductive premise.” 
In other words, it is a necessary truth that the number of 
possible worlds that satisfy the inductive premise and have 
some explanation is greater than the number of worlds that 
satisfy the inductive premise and do not. In fact, he goes 
on to claim that these latter worlds “are quite rare and 
unlikely within the total class of possible worlds” (209). 
Why should we believe this? First, it seems clear that 
the number of possible worlds is infinite. How are we to 
understand Bonjour’s claim about the relative rarity of 
some types of world vis-à-vis others? He concedes that he 
is assuming that it is possible to make sense of the relative 
sizes of classes of possible worlds even when the classes 
are themselves infinite, and cites as an intuitive example, 
Cantor notwithstanding, that there are twice as many 
positive integers as even integers. Despite his faith in the 
“intuitive credentials” of such a claim, we remain unper-
suaded.  
Furthermore, it seems easy to show that among worlds 
in which the inductive premise is satisfied, there are at 
least as many worlds in which there are no “sane” 
objective regularities as there are worlds in which there 
are. Consider a world W in which such regularities hold. 
                                                     
4 Howson, in 2000 (which has greatly influenced our views on induction), pp. 
42-43 considers and nicely rebuts the view that we should reject many of the 
crazy hypotheses because “these concocted alternatives did not predict the 
data independently …”  
Bonjour’s A Priori Justification of Induction - John Meixner and Gary Fuller 
 
 
 229
For each such world there is some other world W* which is 
exactly like W except that in some remote unobserved 
region of W* conditions obtain so as to render false the 
conclusions of all “sane” standard inductions.5 So much for 
the rarity of such “non-explanation” worlds. 
Second, even if Bonjour could show that there are more 
possible worlds in which some explanation holds, it would 
follow that a given world is more likely to be such a world 
only if we assume that each possible world is equally prob-
able. But this is impossible, since any finite number times 
infinity will be larger than one.6 We cannot assign all 
possible worlds equal prior probability, and any attempt to 
make an alternative assignment (e.g., lower probabilities to 
the “crazy” worlds mentioned earlier) will necessarily 
require additional assumptions, which may well turn out to 
be empirical. 
But the situation is even worse than this. The number of 
possible worlds is clearly non-denumerable. Howson 
(2000: 75) has shown that in a non-denumerable possibil-
ity space, a non-denumerable number of the possibilities 
must be assigned a prior probability of zero. Which pos-
sible worlds are to receive that honor? Bonjour gives us no 
clue about this.  
Bonjour’s fresh attempt at an a priori solution to the 
problem of induction, then, is well worth exploring. Sadly, it 
cannot be counted a success: there are too many issues 
left unresolved.  
                                                     
5 We are indebted to Chase Wren (2000) for this suggestion, which we are 
paraphrasing almost verbatim. 
6 This principle, despite being obvious, follows from Archimedes’s Axiom, 
which claims that any for any non-zero real number x, there is a finite number 
of times x can be added to itself such that the resulting sum will exceed any 
finite number specified. 
References  
Bonjour, L. 1998 In Defense of Pure Reason. Cambridge, UK: 
Cambridge University Press. 
Bonjour, L. 2002 Epistemology. Lanham, Maryland: Rowman & 
Littlefield. 
Howson, C. 2000 Hume’s Problem. Oxford: Clarendon Press. 
Mackie, J.L. 1979 “A Defence of Induction” in G. F. MacDonald 
(ed.), Perception and Identity: Essays Presented to A.J. Ayer. 
Ithaca, NY: Cornell University Press. 
Swinburne, R. 2001 Epistemic Justification. Oxford: Clarendon 
Press. 
Wrenn, C. 2000 “Inter-world Probabilities and the Problem of 
Induction.” Unpublished paper delivered at the Central States 
Philosophical Association Conference, Lincoln, NE, Fall 2000. 


Bonjour"s A Priori Justification of Induction

Abstract

Similar works

Full text

Available Versions

Elektronisch archivierte Theorie - Sammelpunkt