On the Justification of Deduction and
Induction

Franz Huber
Department of Philosophy

Institute for the History and Philosophy of Science and Technology
University of Toronto

Penultimate version: please cite the paper that is forthcoming in the
European Journal for the Philosophy of Science

Contents
1 Introduction 3

2 Hume’s Argument against Induction 4

3 Carnap’s Rationalist View of Rationality 7

4 The Continuum of Carnap’s Rationalism 13

5 Goodman’s Fiction about the Facts of Forecast 16

6 Reichenbach Rules 20

7 Haack’s Argument against Deduction 27

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Abstract

The thesis of this paper is that we can justify induction deductively rela-
tive to one end, and deduction inductively relative to a different end. I will
begin by presenting a contemporary variant of Hume (1739; 1748)’s argu-
ment for the thesis that we cannot justify the principle of induction. Then I
will criticize the responses the resulting problem of induction has received
by Carnap (1963; 1968) and Goodman (1954), as well as praise Reichenbach
(1938; 1940)’s approach.

Some of these authors compare induction to deduction. Haack (1976)
compares deduction to induction, and I will critically discuss her argument
for the thesis that we cannot justify the principles of deduction next. In
concluding I will defend the thesis that we can justify induction deductively
relative to one end, and deduction inductively relative to a different end, and
that we can do so in a non-circular way. Along the way I will show how we
can understand deductive and inductive logic as normative theories, and I
will briefly sketch an argument to the effect that there are only hypothetical,
but no categorical imperatives.

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1 Introduction
The thesis of this paper is that we can justify induction deductively relative to
one end, and deduction inductively relative to a different end. Here is how I will
arrive at this unorthodox conclusion. I will first present a contemporary variant of
Hume (1739; 1748)’s argument for the thesis that we cannot justify the principle
of induction. In doing so I will stress that the resulting problem of induction is
problematic only insofar as we have cognitive limitations and ends. (I assume
ultimate cognitive ends to be intrinsic cognitive desires, but nothing in this paper
hinges on this assumption.) We will not solve the problem of induction if we do
not pay due respect to our cognitive limitations and ends, as it is only because of
the latter that the problem of induction is a problem in the first place.

I will then criticize the responses the problem of induction has received by
Carnap (1963; 1968) and Goodman (1954). Logic, whether it is deductive or
inductive logic, can be understood as a normative theory. The upshot of the dis-
cussion of Carnap’s work will be that we get the justification of our normative
theories wrong if we have the wrong theory of normativity. The discussion of
Goodman’s work will drive home the point that we get the inductive case wrong
if we get the deductive case wrong. This critical discussion of the literature will
turn into a constructive proposal once we have incorporated Reichenbach (1938;
1940)’s insight that the justification of norms or imperatives is relative. Logic, like
every (set of) norm(s) or imperative(s), is hypothetical in nature: its validity con-
sists in being the means to attaining some cognitive end. Which logic we ought
to follow depends on what our cognitive limitations and ends are. If we have no
cognitive ends to begin with, we need not obey any logic, and the problem of the
justification of logic – whether it is inductive or deductive logic – vanishes. If we
have cognitive ends, then we need to make these ends explicit before we can even
meaningfully ask whether a given logic is justified relative to these ends.

Carnap and Goodman compare induction to deduction. Haack (1976) com-
pares deduction to induction. I will critically discuss her argument for the thesis
that we cannot justify the principles of deduction next. I will conclude by defend-
ing the thesis that induction is justified by a deductively valid argument relative
to one cognitive end, and deduction is justified by an inductively strong argument
relative to a different cognitive end. Since the two cognitive ends relative to which
we justify induction and deduction, respectively, are different, the justification of
each of these two logical systems is non-circular. It is not claimed that the con-
junction of deduction and induction is justified (by a deductively valid, inductively
strong, or any other argument) relative to the conjunction of these two ends.

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2 Hume’s Argument against Induction
Here is a contemporary variant of Hume’s (1739; 1748) argument for the thesis
that we cannot to justify the principle of induction, whatever its precise form.

Premise 1: we can justify the principle of induction only if there is a de-
ductively valid or inductively strong argument which does not presuppose
its conclusion, whose premises are restricted to information we have, and
whose conclusion says that the principle of induction usually leads from
true premises to true conclusions.

Premise 2: there is no deductively valid argument which does not presup-
pose its conclusion, whose premises are restricted to information we have,
and whose conclusion says that the principle of induction usually leads from
true premises to true conclusions.

Premise 3: there is no inductively strong argument which does not presup-
pose its conclusion, whose premises are restricted to information we have,
and whose conclusion says that the principle of induction usually leads from
true premises to true conclusions.

Conclusion: we cannot justify the principle of induction.

This argument is logically valid. Let us briefly discuss its premises, and the ingre-
dients of the latter. The principle of induction can be characterized as a method
for answering questions to which one does not have the answer in light of infor-
mation that one has. While in general overly simplistic, for the purposes of this
paper the following principle of universal induction will do: from the premise that
all objects about which one has enough information – i.e. whether they are F and
whether they are G – are G if they are F, one may and ought to infer the conclusion
that all Fs are G. Here ‘F’ and ‘G’ are arbitrary predicates. An application of the
principle of induction is an inductively strong argument. The numbers alluded to
in the notion of inductive strength will not play a role in this paper.

Different philosophers will have different views on what precisely is infor-
mation we have. Some might say the information we have are the sentences or
propositions we know, or perhaps assume, to be true. Others might say the infor-
mation we have are the sentences, or perhaps the true sentences, we believe to be
true. As long as there is some information that we do not have, i.e. some ques-
tion to which we do not have the answer, it does not matter much which of these
options or their combinations, if any, we choose (see, however, the caveat below).

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The conclusion of the sub-argument supposedly justifying induction says that
the principle of induction usually leads from true premises to true conclusions. If
there are infinitely many instances of the principle of induction, one needs to make
some assumption (such as defining a measure over the infinitely many instances)
to make sense of the claim that the principle of induction usually leads from true
premises to true conclusions. I will suppose that this assumption is such that the
information we presently have is not sufficient to determine already now that the
principle of induction usually leads from true premises to true conclusions, even
if all its future instances with true premises have a false conclusion.

Premise 3 is true, or so I will presume. It is so because any inductively strong
argument for the conclusion that the principle of induction usually leads from true
premises to true conclusions presumably has to be inductively strong in the sense
of this very principle. Hence any such argument presupposes that the principle
of induction usually leads from true premises to true conclusions. Yet this is pre-
cisely the conclusion we want to derive. So any inductively strong argument for
the conclusion that the principle of induction usually leads from true premises to
true conclusions presupposes its conclusion: it is circular. Therefore there is no
inductively strong argument whose conclusion says that the principle of induction
usually leads from true premises to true conclusions and which does not presup-
pose its conclusion (and hence no such argument whose premises are restricted to
information we have).

The presumption that the inductively strong argument for the principle of in-
duction has to be inductively strong in the sense of this very principle, and with
it premise 3, need not be true if there are several principles of induction that can
inductively justify each other (in each case with respect to the conclusion that
the principle of induction under consideration usually leads from true premises
to true conclusions): the first justifies the second which justifies the third which
justifies the first. If there are only finitely many principles of induction, we are
still caught in a circle. If there are infinitely many principles of induction, we
need not be caught in circle. In this case other arguments are needed to establish
the third premise. It is perhaps worth pointing out that the mere fact that we rely
on infinitely many principles of induction in itself is not reason enough to despair.
However, the burden of proof is with the defender of the inductive justification of
induction, and the infinitist options that come to mind quickly run into problems
(see Skyrms 2000: ch. 3).

Premise 2 is true as long as there is some information that we do not have, and
as long as the assumption made to give precise meaning to the term ‘usually’ does
not violate the above supposition. This is so because every question to which we

5



do not have the answer is, by definition, a question whose answer requires infor-
mation which we do not have. The principle of induction is a general principle that
applies to all questions, including those to which we do not have the answer. And
it is a truth of logic that there is no deductively valid argument whose premises
are restricted to information we have, and whose conclusion says something that
is the answer to a question to which we do not have the answer. Therefore, as long
as there is some information that we do not have, and the supposition mentioned
above is met, there is no deductively valid argument whose premises are restricted
to information we have, and whose conclusion says that the principle of induction
usually leads from true premises to true conclusions (and hence no such argument
which does not presuppose its conclusion).

Two points are worth noting. First, the problem of induction does not arise
for omniscient beings who have as information all truths. If it is indeed the case
that the principle of induction usually leads from true premises to true conclu-
sions, then these beings have this information. And if it is not the case that the
principle of induction usually leads from true premises to true conclusions, then
these beings have this information. Either way there is a deductively valid argu-
ment which does not presuppose its conclusion, whose premises are restricted to
information these beings have, and whose conclusion says that the principle of
induction usually leads from true premises to true conclusions – or does not do so,
as the case may be. We who are not omniscient cannot provide such a deductively
valid argument, because we do not have enough information (here the choice of
what we take to be information that we have matters somewhat). The problem of
induction is our problem, not that of omniscient beings.

Second, the problem of induction vanishes for cognitively contented beings
who have no curiosity or other cognitive ends. The problem of induction vanishes
for such beings in the sense that, while perhaps still a problem, it is one that need
not, and does not, bother such contented beings. For us who are not (cognitively)
contented in this way the problem of induction is real and does not vanish, because
we generally want to have the information that we do not have – and often we need
to in order to survive. Curiosity may have killed the cat. However, the cat would
have died much sooner if it had not been curious.1

1To be sure, we do not need to have information about how induction works, or if it can be
justified, in order to apply induction and be good at doing so. However, we do need to have such
information if we want to delegate the task of making inductive inferences to computers. Besides
our philosophical curiosity I take this engineering task to be the major reason for the interest in
the justification of induction. To the extent that philosophy is a normative discipline it is a branch
of engineering (cf. Quine 1986).

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Premise 1 is false. However, the premise 1 is not false because it makes strong
internalist assumptions about justification. I happily grant these. Rather, premise
1 is false, because we can justify the principle of induction by a deductively valid
argument which does not presuppose its conclusion, whose premises are restricted
to information we have, but whose conclusion says something other than that the
principle of induction usually leads from true premises to true conclusions. We
will see in section 6 what this other feature is relative to which we can justify the
principle of universal induction by a deductively valid argument.

3 Carnap’s Rationalist View of Rationality
Carnap (1934; 1936; 1937; 1947) famously denied Kant (1902)’s dictum (from
The Critique of Pure Reason) that there are synthetic truths that we can know a
priori. Afterwards Carnap worked extensively on probability and inductive logic
(Carnap 1950; 1952; 1962). However, with the possible exception of Carnap
(1962: §41) it is only in response to Nagel (1963: 820f) and, especially, Kemeny
(1963) and Burks (1963) that Carnap (1963b: 978) briefly discusses “the contro-
versial problem of the justification of induction.”

For Carnap the principle of induction takes the form of an inductive logic.
Hence for him the justification of induction boils down to justifying the axioms
of his inductive logic. These axioms specify a confirmation function (in Carnap
1950 it is one single confirmation function; in Carnap 1952 it is an entire family
of confirmation functions). These axioms, and their logical consequences, are
analytically true, as Carnap stresses frequently (for instance, Carnap 1963a: 71f).

According to Carnap (1963b: 987) we should “understand the problem of the
justification of induction as the question as to what kinds of reasons can be given
for accepting the axioms of inductive logic.” These reasons take the form of “con-
ditions of adequacy” (Carnap 1963b: 977). They “are based upon our intuitive
judgments concerning inductive validity, i.e., concerning inductive rationality of
practical decisions (e.g., about bets)” (Carnap 1963b: 978). From this he infers
that “[i]t is impossible to give a purely deductive justification of induction,” and
that these “reasons are a priori.”

The notion of analyticity defended by Carnap (1934; 1936; 1937) defines ana-
lytic truth syntactically as derivability in a logical system. Carnap (1947) defines
analytic truth semantically as validity in a family of models. This difference does
not matter for us (unless the “reasons” mentioned above are semantically valid,
but not derivable syntactically, and there is no indication that this is so).

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What does matter for us is that Carnap’s claim that “[i]t is impossible to give
a purely deductive justification of induction” implies, on his very own account of
analyticity, that the “reasons” which we cite in support of the various axioms of in-
ductive logic, and which are “based upon our intuitive judgments about inductive
validity,” are synthetic. It is important to note that this is not true of what these
reasons are reasons for, namely the analytically true axioms of inductive logic.
Since Carnap (1963b) also claims that these “reasons are a priori,” it follows that
the reasons, or conditions of adequacy, that justify us in accepting the axioms
of inductive logic are synthetic truths that we can know a priori. The source of
our a priori knowledge of these synthetic reasons, or conditions of adequacy, is
“intuitive judgment.”

This is a curious thing to say, especially for Carnap, who, in Carnap (1934;
1936; 1937; 1947), held that there are no synthetic truths that we can know a pri-
ori. Let us have a closer look to see what has gone wrong. Carnap follows Kemeny
in regarding the following condition of adequacy to be “intuitively valid” (Carnap
1963b: 977): the confirmation function “c must define a system of bets that is
[strictly] fair.” (Kemeny 1963: 720) This condition of adequacy is the reason why
we ought to accept the axioms of probability theory as axioms of inductive logic.
The axioms of probability theory themselves are analytically true sentences of the
object language (and presumably we can know them a priori). In contrast to this
the condition of adequacy is a synthetic sentence of the metalanguage. Yet we
can still know it a priori, because “questions of rationality are purely a priori”
(Carnap 1963b: 983). In my opinion it is here, in his view of rationality, where
things go wrong. Before explaining why I think so, let me briefly discuss why
Carnap might have thought that “questions of rationality are purely a priori”. He
might have thought so because of another dictum of Kant (1902)’s (this time from
The Groundwork of the Metaphysics of Morals), namely the idea that there are
absolute norms or categorical imperatives.

On the view of rationality, or normativity, adopted in this paper there are no
categorical imperatives. Needless to say, there are utterances of normative, i.e.
prescriptive or evaluative, claims whose grammatical form is categorical. Indeed,
I will give an example of such a grammatically categorical imperative momentar-
ily. What the present view denies is that there is any categorical deontic reality
that is referred to by such grammatically categorical imperatives. On the present
view of normativity there are only means to ends. Means-end relationships in-
clude being a contributing cause and other potentially non-deterministic causal
relationships that cannot be established by formal methods alone, as well as being
logically equivalent and other logical relationships that can be so established.

8



Which ultimate ends we have is a factual matter. Which actions, or intentions
to act, are means to which ends often is so as well. What is never a factual matter
on the present view is which ultimate ends one ought to have. In what follows I
will briefly sketch an argument for this view.

Some imperatives are hypothetical imperatives. Say that Emmy should (intend
or will to) not smoke. The grammatical form of this imperative is categorical. It is
valid, because not smoking is a means to attaining the end of not developing lung
cancer, an end Emmy is hypothetically assumed to have. Now enter Occam’s
razor, which is itself a norm or imperative. Occam’s razor urges us to not mul-
tiply entities without necessity. In particular, it requires us to avoid additionally
postulating the existence of a second kind of imperative, the category of categor-
ical imperatives – unless we have to. We do not have to, though. Even if there
were imperatives which we did not yet understand as hypothetical imperatives, it
would not follow that we would have to postulate a second category of categorical
imperatives. Instead we could always try harder. Occam’s razor in turn is not a
categorical imperative, but a means to attaining the truth efficiently (Kelly 2007).
And it just so happens that I desire to hold true and informative beliefs, and to
arrive at them efficiently.

Of course, things would be radically different if there was an argument to the
effect that there are categorical imperatives. That, however, is not even possible,
for we will see that the principles of logic – any logic, that is – are hypothetical
imperatives, and no argument can do without some logical principle or other.

This is why I do not believe in categorical imperatives. To the extent that you
also aim at holding true and informative beliefs, and to arrive at them efficiently,
you ought to do so as well. Not because I tell you so, but because doing so is a
means to attaining your ends. (This is not to say that holding false beliefs cannot
be a means to attaining ends other than holding true beliefs.)

Korsgaard (2008) argues that there can be no requirement to take the means
to our ends unless there are also required ends. Let me briefly point out why her
arguments do not undermine the above claim. First, all that Korsgaard (2008)
shows is that there can be no categorical requirement to take some action unless
there are also ends that are categorically required. She does not show that there
can be no hypothetical requirement that is conditional on some end that itself is
not required. Someone who denies the existence of categorical imperatives will
expect and embrace this conclusion rather than conclude that there are ends that
are categorically required: that there are only hypothetical imperatives, and no
action that is categorically required, is what the denier of categorical imperatives
– Korsgaard (2008)’s “empiricist”– has claimed all along!

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Second, contrary to what Korsgaard (2008) claims the empiricist cannot only
explain why the instrumental principle that one ought to take the means to one’s
ends can motivate us. That is done by the ends or desires a hypothetical imperative
is conditional upon. She can also explain why it can guide us. In particular, we
can violate the instrumental principle in two ways: by having false or incomplete
beliefs about what the means to our ends are, and about the consequences of taking
these means; as well as by failing to intend or will to take the means to our ends.
What ends we have is not always transparent to us. Sometimes this is so, because
we make mistakes and have false beliefs about the consequences of realizing, i.e.
taking the means to attaining, our ends. Sometimes this is so, because we simply
cannot foresee all the consequences of realizing our ends. This is true for ends
that are means to other ends. It is also true for ultimate ends (if we can have more
than one), as we may miss that the only way of realizing one ultimate end prevents
us from realizing another ultimate end that is at least as important to us.

Korsgaard (2008: 10f) thinks the empiricist must hold that there are no, and
cannot be any, violations of the instrumental principle. She considers an agent
whose ends are “shaped by [...] terror, idleness, shyness or depression” (Korsgaard
2008: 12) and thinks the empiricist cannot hold that this agent acts irrationally.
However, the beliefs (about the pains and pleasures that are actually provided
by taking the means to her ends) that this agent acts upon are simply false. It is
precisely because this agent could or even does have other, true beliefs, yet fails to
act upon the latter, that we consider her to act irrationally when she lets depression
stop her from working (she acts as if she believed the work was difficult when she
could or even does have the other, true belief that it is not), shyness stop her from
calling (she acts as if she believed calling was embarrassing when she could or
even does have the other, true belief that it is not), idleness stop her from going
downtown (she acts as if she believed going was inconvenient when she could or
even does have the other, true belief that it is not), and terror stop her from riding
the roller coaster (she acts as if she believed riding was dangerous when she could
or even does have the other, true belief that it is not).

To be sure, this is different for omniscient agents who should take precisely
these actions that they intend or will to take, unless they suffer from weakness of
will. Furthermore, omnipotent agents intend or will to take precisely these actions
that they do take. So agents who are omniscient and omnipotent and who do not
suffer from weakness of will can, indeed, not violate the instrumental principle.
Nor can such agents be guided by the instrumental principle. However, this is as
it should be: omniscient and omnipotent agents who do not suffer from weakness
of will do exactly what they should do. They have no need for any guidance.

10



Finally, the Ought in the instrumental principle that one ought to take the
means to one’s ends is not derived (Korsgaard 2008: 6f) from the Is of having
some end. Rather, the Ought is conditional upon the Is of having some end. Hy-
pothetical imperatives are not categorical imperatives that some hypothetical, or
conditional, be obeyed. Nor are they hypotheticals, or conditionals, whose con-
sequent ought to be obeyed if its antecedent obtains (or is obeyed). Hypothetical
imperatives are a sui generis category, somewhat similar to conditional beliefs
which also cannot be reduced to beliefs that some conditional is true, or some
conditional whose consequent is believed if its antecedent obtains (or is believed).

Hypothetical imperatives or norms are great. For instance, they allow us to ex-
plain why norms have normative force. They have such force, because ultimately
they are conditional upon ultimate ends – that is, subjective intrinsic desires – and
desires have force, ranging from mere wants to genuine needs. In the purely the-
oretical domain of epistemology, the strength, or force, of our cognitive desires
explains the normative force of our doxastic or cognitive norms. In this domain
taking the means to our cognitive ends can come in conflict only with taking the
means to other cognitive ends of ours, as we never leave the internal world that
is our mind. Outside the purely theoretical arena, where your taking the means
to your ends may conflict with my taking the means to my ends, things become
more difficult, but not different: there are no required means without some de-
sired ends. In contrast to this, we presumably need to postulate the existence of a
special authority or deity, or of moral facts or real virtues or objective intrinsic val-
ues, or other ontologically dubious categories in order to explain how categorical
imperatives or norms can have normative force.

However, which hypothetical imperatives hold, and which desires they are ul-
timately conditional upon, is not, in general, an a priori matter. First, it is often an
empirical question which desires we happen to have: Emmy might desire to de-
velop lung cancer, she just happens not to. For her this might be an a priori matter,
but for us it is not; and sometimes not even our own desires are a priori matters
for us. Second, it is often an empirical question which means-end relationships
obtain: that not smoking is a (non-deterministic causal) means to avoiding the
development of lung cancer is an empirical, a posteriori discovery. In contrast to
this, categorical imperatives do not depend on empirical matters that can only be
discovered a posteriori. At least this is so if they are “grounded” in moral facts or
other ontologically dubious categories that are not subject to empirical investiga-
tion in the way means-end relationships are.

Since Carnap (1963b: 983) holds that “questions of rationality are purely a
priori,” he cannot understand norms as hypothetical imperatives. Assuming that

11



there is no third category of imperatives, it follows that Carnap must understand
norms as categorical imperatives. Since the validity, or invalidity, of categorical
imperatives cannot, in general, be discovered empirically, it is then only natural
for him to postulate a powerful source of knowledge such as “rational insight” or
“intuitive judgment” that allegedly enables us to know a priori which categorical
imperatives hold.

Unfortunately philosophical problems are not solved by merely postulating
solutions to them, no matter how many times we call them “rational.” That’s
not engaging with a problem, that’s just giving up and not admitting it. Carnap,
despite his empiricist confessions (Carnap & Hahn & Neurath 1929), seems to
have indulged in just such a rationalist view of rationality. He seems to have
thought of the conditions of adequacy as categorical imperatives: “c must define
a system of bets that is [strictly] fair”. This, so it seems to be Carnap’s view, is a
categorical imperative about how we should use the term ‘degree of confirmation,’
about how to explicate ‘degree of confirmation.’ On this view, we find out about
the validity of this categorical imperative by intuitive judgment, which enables us
to know a priori that the condition of adequacy holds.

Carnap’s curious view results because he seems to have thought of these con-
ditions of adequacy as categorical imperatives of explication which justify the an-
alytic truth of the axioms of inductive logic. In contrast to this, the view adopted
in this paper holds that the axioms of inductive logic are hypothetical imperatives
of belief that we justify by showing, i.e. providing an argument for the conclusion
that, they are the means to attaining the cognitive end they are conditional upon.

Specifically, Carnap seems to have thought that conditions of adequacy are
categorical imperatives that are formulated in the metalanguage and that tell us
how we ought to use words in the object language. They thus take the form: it
ought to be the case that C, O (C). The axioms of inductive logic are analytic
truths in the object language of the form: A. The conditions of adequacy regulate
the use of words in the object language. They are the reasons why we ought to
accept the axioms. We thus get the following picture: O (C) → A, where → is the
reason (for acceptance) relationship between synthetic norms in the metalanguage
and analytic truths in the object language.

On the present view the principle of induction and the axioms of inductive
logic are not categorical imperatives to use words in certain ways, but hypothetical
imperatives for organizing one’s beliefs B at a given moment in time. We justify
these hypothetical imperatives to believe in a certain way by showing or arguing
that they are the means to attaining the cognitive end E they are conditional upon.
On the present view, the picture is this: OE (B).

12



Which ultimate cognitive ends E we have is a factual question, or a matter of
stipulation if we consider how we should believe if we had certain cognitive ends.
The same is true for what we actually believe. Which means-end relationships
obtain between one’s believing, or intending to believe, B, and various ends E
sometimes is an a priori matter, if theorem proving is – though not always, as we
have seen above. For instance, to the extent that theorem proving is an a priori
matter2, we will see in section 5 that it can sometimes be discovered a priori
that B-ing is a necessary, or a necessary and sufficient, condition for attaining E.
(Whether this is also true for the mere intention to B depends on whether it is
a priori that intentions are necessary for actions.) However, when the means-end
relationship can be so established, it is by honest theorem proving, not by intuition
mongering, rational or otherwise.3

4 The Continuum of Carnap’s Rationalism
It is natural to wonder if Carnap was aware that his view commits him to a multi-
tude of synthetic truths that we can know a priori, something whose existence he
had denied earlier. This does not seem to be the case.

2I do not assume that theorem proving is an a priori matter, and nothing in this paper depends
on the assumption that means-end relationships can be established a priori.

3There is another reason why Carnapian explication requires intuitive judgment to be a pow-
erful source of a priori knowledge. According to Carnap (1962: 3) an explication is “the trans-
formation of an inexact, prescientific concept, the explicandum, into a new exact concept, the
explicatum.” An explicatum is to satisfy four requirements: similarity (in meaning), exactness,
fruitfulness, and simplicity. “The explicatum is to be similar to the explicandum in such a way
that, in most cases in which the explicandum has so far been used, the explicatum can be used;
however, close similarity is not required, and considerable differences are permitted” (Carnap
1962: 7). In this respect explication differs from analysis, where the analysans is to be identi-
cal (in meaning) to the analysandum in such a way that, in all cases in which the analysandum
has been used so far, the analysans can be used; strict identity (in meaning) is required, and no
differences are permitted.

Explication requires intuitive judgment to be a powerful source of a priori knowledge, because
the method of counterexamples does not work for explication. We can grant that Gettier (1963)
shows that justified true belief is not identical in meaning to knowledge. What Gettier (1963)
does not show is that justified true belief is not similar in meaning to knowledge, because a few
counterexamples to the identity of meaning do not establish gross dissimilarity. Intuitive judg-
ment already has to be a powerful source of knowledge if it can yield a priori knowledge that
analysans and analysandum are not identical (in meaning). It has to be an even more powerful
source of knowledge if it can yield a priori knowledge of gross dissimilarity (in meaning) between
explicatum and explicandum.

13



In Carnap (1968) and in the unpublished manuscript “How can induction be
justified?” in which he responds to Lenz (1956), and which seems to have served
as a precursor of Carnap (1968), Carnap makes it very clear that he thinks he
does not rely on any synthetic principles.4 Carnap distinguishes between two
problems of justification. The second problem arises only if there is more than
one confirmation function that satisfies the axioms of inductive logic, as is the
case for the system presented in Carnap (1952). It is a problem of riches, namely
the problem of justifying the choice of one of these confirmation functions over the
others. The first problem is the problem of justifying the axioms of inductive logic
themselves. It is this problem that corresponds to the problem of the justification
of induction in Carnap’s work, and that is of relevance to the present paper. The
story that Carnap (1968) tells is the same story as above, except that the reasons
or conditions of adequacy are now called “requirements of rationality” (Carnap
1968: 260), and that the source of our a priori knowledge of these requirements
of rationality is called “inductive intuition” (Carnap 1968: 265).5

Here is what Carnap (1968: 264) says about these requirements of rationality:

Now let us think about the question of the kind of factors we may
legitimately appeal to in such arguments about rationality. As a matter
of fact, in the history of the subject the following factors have often
been used by one or the other philosopher:

(a) deductive logic or deductive reasoning;

(b) inductive reasoning;

(c) past experience;

(d) general synthetic principles (as for instance the principle of the
uniformity of the world).

To save time, I will not speak about the last point, because I believe
that today the majority of philosophers reject this, though some, for
instance Russell, still appeal to such a principle.

Apparently Carnap is of the opinion that he does not rely on general synthetic
principles (d). However, his very own definition of analyticity from the 1930s and

4I am grateful to André W. Carus for pointing me to, and providing me with, these three papers.
5In the unpublished manuscript Carnap also talks about “inductive inertia”, which plays a role

for the second problem of riches, and which seems to be similar to what contemporary decision
theorists call “risk” (Buchak 2014).

14



1940s, and his claim that “it is impossible to give a purely deductive justification
of induction” from the 1960s imply that he does rely on such general synthetic
principles. This will be clear once we recall that, for Carnap, inductive reasoning
does not really differ from deductive reasoning, as both (a) and (b) consist of
analytic truths.

Carnap (1968: 265) goes on to explain that both (a) and (b) are “legitimate”
and “indispensable”, and “that the situation in deductive logic is exactly the same”:
just as “deductive intuition” (Carnap 1968: 266) is the source of our a priori
knowledge of the validity of deductive reasoning, so inductive intuition is the
source of our a priori knowledge of the validity of inductive reasoning. Carnap
(1968: 265) is aware that this “looks like defending a vicious circle,” but he

believe[s] all procedures of self-clarification, of making clear to our-
selves what it is we have in mind, are in a way circular. We clarify B
through A, and then we turn around and explain A with the help of B.
I think we cannot do without it. If a person were unable to distinguish
valid from invalid steps in inductive reasoning, even in the simplest
cases, in other words, if he were inductively blind, then it would be
hopeless to try to convince him of anything in inductive logic.

This quote makes clear that the problem Carnap is concerned with is how we can
justify the results of explication6, of “making clear to ourselves what it is we have
in mind.” Carnap is concerned with the justification of the axioms of (deductive
and) inductive logic which govern the use of the term ‘degree of confirmation.’
In other words, Carnap is concerned with the justification of how we should talk
about believing or reasoning. This is in sharp contrast with what I interpret Hume
(1739; 1748) to be concerned with, and with what I am concerned with, namely
the justification of (deductive and) inductive logic itself: the question why we
should believe or reason in the way (deductive and) inductive logic prescribes.

Much like Carnap I will conclude in sections 6 and 7 that induction and de-
duction can be justified. However, in contrast to Carnap, I do not think that we
can justify deduction and induction circularly, and by appeals to intuition in the
way he has envisaged.

6Carnap (1962: 172f) distinguishes between explication and clarification. The latter is the
process of clarifying the explicandum before it is explicated by an explicatum in explication.
While the definition in an explication must not be circular, clarification indispensably is. I fail
to see how clarification can be separated from explication, and Carnap (1968) does not seem to
put much emphasis on this distinction anymore either.

15



In part this difference may be due to me thinking of deductive and inductive
logic as normative theories regulating how we should believe or reason rather than
thinking of the requirements of rationality as regulating how we should talk about
believing or reasoning. In part this difference may also be due to me ascribing
at best heuristic value to deductive and inductive intuition. Mainly, though, this
difference is due to me not believing in categorical imperatives. In my opinion
Carnap’s view of rationality, or normativity, is wrong. Since he has the wrong
theory of normativity, it is not surprising that Carnap also gets wrong the justifi-
cation of normative theories, such as deductive and inductive logic.

5 Goodman’s Fiction about the Facts of Forecast
Goodman (1946; 1947; 1954) is very critical of Carnap’s syntactic program. The
relevant section of the third chapter of Fact, Fiction, Forecast (1954) on “The New
Riddle of Induction” is one of the best known pieces of 20th century philosophy
of induction (see Stalker 1994).

Interestingly, though, the views expressed by Goodman in this chapter are very
similar to Carnap’s views on explication. Goodman follows Carnap in comparing
inductive logic to deductive logic. He thinks that a particular deductive inference
such as

Athens is the capital of Greece. Therefore, there is a capital of Greece.

is justified by conforming to the valid general rules of deductive logic, including

F (a) ∴∃xF (x)

The general rules of deductive logic in turn are justified by conforming to accepted
deductive practice. Goodman (1983: 64) is, of course, aware that

[t]his looks flagrantly circular. ... But this circle is a virtuous one.
The point is that rules and particular inferences alike are justified by
being brought into agreement with each other. A rule is amended
if it yields an inference we are unwilling to accept; an inference is
rejected if it violates a rule we are unwilling to amend. The process of
justification is the delicate one of making mutual adjustments between
rules and accepted inferences; and in the agreement achieved lies the
only justification needed for either.

16



At least to my ears this sounds very similar to Carnap (1968: 265)’s defense of
the circular nature of “self-clarification.” It also does not work. More specifically,
it does not work for the deductive case, nor does it work for the inductive case.7

It is simply a fiction that it would be a fact that there is agreement between the
particular deductive inferences we accept, and the general rules of deduction that
are valid. Many do not accept particular deductive inferences such as

Athens is the capital of Greece. Therefore, Athens is the capital of
Greece or the moon is made of cheese.

This particular deductive inference is not accepted by many of us, I claim, though
only few have attempted to amend the following general rule of deduction that is
valid (in classical logic) and that the particular deductive inference conforms to:

A ∴ A ∨ C

Many more examples can be given, even if we bracket the material conditional
whose truth conditions presumably have caused incredulous stares on the faces
of most of us when we first encountered them. Indeed, far from there being any
agreement, the discrepancy between the particular deductive inferences we accept,
and the general rules of deduction that are valid, is so large that some philosophers
have even felt the need for supplementary theories to bridge the gap. Grice (1975)
proposes a supplementary theory according to which the maxims of quantity and
of relevance and of manner that require us to be informative and relevant and per-
spicuous, respectively, explain why many do not accept the particular deductive
inference above, even though it conforms to a general rule of deductive logic that
is valid. The latter rules matter mainly for the maxim of quality according to
which we should tell the truth. Jackson (1979) proposes a supplementary theory
that attempts to explain the gap between particular accepted inferences involving
indicative conditionals and general valid rules of deductive logic involving the
material conditional. Lewis (1986: fn 6) adopts this theory. Apparently all these
philosophers agree that the discrepancy between accepted deductive inferences
and valid deductive rules is so large that a supplementary theory is needed.

The problems Goodman’s view has with the deductive case carry over to the
inductive case. Indeed, the very difference between deductive and inductive logic
seems to vanish on Goodman’s view. We have seen that the difference between
deductive and inductive reasoning vanishes for Carnap: both his (a) and (b) consist
of analytic truths.

7The same is true of Strawson (1952: ch. 9; 1958)’s attempt to define away the problem of
induction.

17



The situation is somewhat similar on Goodman’s view. There are particular
inferences some of us accept, and other particular inferences those same of us do
not accept. On Goodman’s view it seems arbitrary to further divide the acceptable
particular inferences into those that are acceptable qua deductive inferences, and
those that are acceptable qua inductive inferences. The inference from Athens
being the capital of Greece to there being a capital of Greece is just as acceptable
as a deductive inference as it is acceptable as an inductive inference. The inference
from Athens being the capital of Greece to Athens being the capital of Greece or
the moon being made of cheese is just as unacceptable as a deductive inference as
it is unacceptable as an inductive inference.8

To be sure, there is a difference between deductive and inductive logic. How-
ever, this difference is not grounded in the source of our logical knowledge, the
difference between deductive and inductive intuition, as Carnap has it. Nor is this
difference grounded in our inferential practice, the difference between inferences
that are acceptable qua deductive inference and inferences that are acceptable qua
inductive inferences, as Goodman will have it. The difference between deductive
and inductive logic is a difference in the cognitive ends these normative systems
are a means to. In this sense this difference is grounded in the normativity of logic.

How do things work in the deductive case? How do we justify the general rules
of deductive logic? We justify them in the exact same way as we justify all norms,
namely by showing or arguing that they are the means to some end (whether this
end is desired by anybody matters only for the relevance of the norm in question,
not its validity). The general rules of deductive logic are justified relative to the
end of reasoning in a way that is truth-preserving with logical necessity. For all
and only those particular inferences that conform to the general rules of deductive
logic are truth-preserving with logical necessity. The epistemological significance
of Gödel (1930)’s correctness and completeness theorems for classical logic lies
in establishing this means-end relationship. The epistemological significance of
Gödel (1931)’s (first) incompleteness theorem lies in establishing that, with log-
ical necessity, we miss out on something regarding the end of holding only true
beliefs, and as many thereof as possible, once we widen our cognitive interests
and become curious about arithmetical questions (and believe an answer to an
arithmetical question just in case we can derive it in Peano arithmetic). For either
not every statement of the language of Peano arithmetic that is derivable in Peano

8Goodman’s positive theory of induction in terms of “entrenchment” is developed in Goodman
(1983: sct. IV). If (inductive) logic is a normative theory it says that we ought to project or infer
inductively what we do in fact project or infer inductively, and thus infers an Ought from an Is.

18



arithmetic is true (in the standard model) – in this case we miss out by not holding
only true beliefs; or not every statement of the language of Peano arithmetic that
is true (in the standard model) is derivable in Peano arithmetic – in this case we
miss out by not holding as many true beliefs as possible.

In section 7 I will turn to the entirely valid objection that this justification of
the rules of deductive logic relative to the end of reasoning in a way that is truth-
preserving with logical necessity is viciously circular, because it uses the rules of
deductive logic. Before doing so, let me briefly give two more examples of how
we justify a system of logic understood as a system of hypothetical imperatives
governing how one should believe or reason relative to a particular cognitive end.

The first example is the non-monotonic logic developed by Kraus & Lehmann
& Magidor (1990). Its rules are justified – not relative to the end of reasoning
in a way that is truth-preserving with logical necessity, but – relative to the end
of reasoning in a normally truth-preserving way: all and only those particular
inferences that obey these rules are such that their conclusion is true in all of
the most normal possible worlds in which all premises are true. Three things
are interesting about this example. First, I think this system has a better chance to
come out as the system of deductive logic on Goodman’s approach, even though it
agrees with classical deductive logic on the examples above. Second, this system
is just as much a deductive logic as it is an inductive logic, thus stressing that
this dichotomy is somewhat arbitrary. Third, this example illustrates (as does
the next) that sometimes it can be discovered a priori (see, however, footnote 2)
that obeying a system of norms is a provably necessary and sufficient means to
attaining a particular end. However, it is by honest theorem proving rather than
by appeals to intuition that we can obtain such a priori knowledge.

The first example illustrates that different cognitive ends give rise to different
logical systems. The next example illustrates that different propositional attitudes
give rise to different logical systems. It is the probability calculus whose rules are
justified relative to the end of avoiding accuracy domination (Joyce 1998; 2009):
all and only those degree of belief functions that obey these obligations are such
that it is not the case that they are accuracy dominated in the sense that there exists
an alternative degree of belief function that is at least as accurate in all possible
worlds, and more accurate in at least one. This example is an instance of epistemic
consequentialism (Percival 2002, Stalnaker 2002, Brössel & Eder & Huber 2013)
according to which epistemic rationality is a species of instrumental rationality. If
we supplement this view with explicitly denying the existence of objective values
so that there is nothing objectively wrong or right about what ultimate cognitive
ends one happens to have we arrive at the view adopted here.

19



The upshot of this is three-fold. First, we need to specify a cognitive end
before we can even meaningfully ask if we can justify the principle of induction
relative to this end. We have already seen that the problem of induction vanishes
for a cognitively contented being. Now we see that different cognitive ends give
rise to different principles of induction, or inductive logics, and different problems
of induction.

Second, we must pay attention to the fact that different propositional attitudes
require different principles of induction. Not all cognitive ends are meaningful for
all propositional attitudes. For instance, someone who has only degrees of belief
might not be able to meaningfully desire the cognitive end of holding only true
beliefs, and as many thereof as possible.

Third, we must not forget that different agents have different cognitive abilities
and limitations. If our doxastic agent is omniscient, she does not need a princi-
ple of induction. If someone sometimes forgets, is not always able to voluntarily
decide what to believe (to a specific degree), or cannot compute everything com-
putable, our principle of induction needs to take these limitations into account.

6 Reichenbach Rules
All this seems to have been clear to Reichenbach (1938; 1940) who was interested
in figuring out the limit of the relative frequency of types of events in sequences
of events. To this end he formulated the straight rule, which is his version of the
principle of induction, his inductive logic. Here is my preferred variant of the
straight rule: from the premise that m out of the n events about which one has the
information whether they are of type A are of type A, one may and ought to infer
the conclusion that the limit of the relative frequency of events of type A in any
sequence of events continuing these n events is m/n.

Whatever one thinks of the end of figuring out limits of relative frequencies of
types of events in sequences of events, it is insightful to consider how Reichenbach
justifies the straight rule relative to this end. He does so by providing a deductively
valid argument which does not presuppose its conclusion, and whose premises
contain enough mathematics to make sense of the numbers that are mentioned in
the straight rule. The conclusion of this deductively valid argument does not say
that the straight rule usually leads from true premises to true conclusions. Instead
it says that obeying the straight rule is a means to attaining the cognitive end of
converging to the true limit of the relative frequency of any type of event in any
sequence of events for which there is such a limit.

20



What Reichenbach shows is that, for any sequence of events s and any type of
event A: the limit of the relative frequency of A in s exists if and only if the straight
rule converges to it. Logicians may want to think of this result as establishing the
correctness and completeness of Reichenbach’s inductive logic with respect to the
cognitive end of converging to the true limit of the relative frequency of any type
of event in any sequence of events for which there is such a limit: provided the
information in the premises is true, the straight rule converges only to true limits
(correctness), and it converges to all true limits (completeness).

One of the objections to Reichenbach’s argument is that the straight rule is not
the only rule that converges to all and only true limits. The rule that conjectures
(m + 5)/n when the straight rule conjectures m/n does so, too. In response to
this objection one may point out that the situation is exactly analogous in deduc-
tive logic. There are many different axiomatizations of classical logic. However,
all these axiomatizations are functionally equivalent in the sense that all of them
characterize the exact same set of particular inferences. This means that, as far
as the cognitive end of reasoning in a way that is truth-preserving with logical
necessity is concerned, all these different axiomatizations are on a par. Note that
these axiomatizations are not on a par with respect to other ends – say, the end of
deriving conclusions in as few steps as possible.

Consider a different example. Drinking a glass of water quenches your thirst
if and only if drinking a glass of water or some other drink does. As far as the
end of quenching your thirst is concerned drinking a glass of water is on a par
with drinking another drink. The two are functionally equivalent with respect to
this end. Of course, there are other ends, such as quenching your thirst with a
sweet drink or a drink with few calories, with respect to which drinking a glass
of water is not on a par with drinking another drink. But that is besides the point,
because that is not claimed. Similarly, the straight rule and the above alternative
are on a par as far as the cognitive end of converging to the true limit of the
relative frequency of any type of event in any sequence of events for which there
is such a limit is concerned. The two are functionally equivalent with respect to
this cognitive end. It is not claimed that these two rules are functionally equivalent
with respect to any other end – say, making true or accurate conjectures about the
type of the next event.

Given that you have the end of quenching your thirst with a drink with few
calories, you ought to drink a glass of water. Yet you cannot just drink a glass of
water. You can only drink a particular glass of water containing a certain number
of H2O molecules at a particular time in a particular manner. However, drinking
another glass of water containing the same number of H2O molecules, but differ-

21



ent ones, at a slightly later time in a slightly different manner would do equally
well: it would be functionally equivalent with respect to the end of quenching
your thirst with a drink with few calories. Therefore, given that you have this
end, what you really ought to do is something that is functionally equivalent to
drinking a particular glass of water containing a certain number of H2O molecules
at a particular time in a particular manner, where functional equivalence is un-
derstood with respect to the end of quenching your thirst with a drink with few
calories. In the same way we cannot just follow the straight(-forward) rule or the
above alternative or some other alternative that converges to the same limits as
the straight rule. We can only follow one of these alternative rules. Therefore,
all that the non-uniqueness objection to Reichenbach’s argument shows is that we
have to formulate its conclusion more carefully as follows: given that we have the
cognitive end of converging to the true limit of the relative frequency of any type
of event in any sequence of events for which there is such a limit, we ought to
obey a rule that is functionally equivalent to the straight rule, where functional
equivalence is understood with respect to this cognitive end.

Another objection is the slogan, associated with Keynes, that “in the long
run we are all dead.” This suggests that nobody has, or should have, the end of
converging to the true limit of the relative frequency of any type of event in any
sequence of events for which there is such a limit. Even if this was the case, the
objection would miss the point. Reichenbach’s argument establishes a means-end
relationship without suggesting that one ought to, or does, have the relevant end.

Furthermore, the situation in deductive logic is again analogous. Just as in the
long run we are all dead, nobody lives in any logically possible world other than
the actual one. Yet this does not prevent people from desiring to have beliefs that
are consistent, i.e. true in some logically possible world that need not be the actual
one. Now, consistency alone may not make beliefs desirable, but think about how
undesirable beliefs are that are not even consistent. Similarly, convergence to
the true limit alone may not make a sequence of conjectures desirable, but think
about how undesirable a sequence of conjectures is that does not even converge
to true limit. It is no coincidence that estimators in statistics are called “strongly
consistent” if their conjectures converge (with probability one, or almost surely).

A third objection is the reference class problem. In fact, there are two of
these problems: one for the type of event A (each event instantiates many types)
and one for the sequence of events s (each event belongs to many sequences).
Picking the narrowest reference class for A (the type whose only instantiation is
the next event) is picking a type that contains too few events, and picking the
broadest reference class for s (the sequence of all events) is picking a sequence

22



that contains too many events. Picking any other reference class is arbitrary.
The reference class problems are indeed problems if one wants to use observed

relative frequencies to determine probabilities, say, the probability that the next
event is of a certain type. However, they are not problems for the straight rule,
because the latter does not attempt to do this. The straight rule does not even
mention probability or the next event. Instead it conjectures the limit of the relative
frequency of any type of event A in any sequence of events that continues the n
events about which one has the information whether they are of type A.

Now, to be sure, Reichenbach was also interested in using observed relative
frequencies to determine the probability that the next event is of a certain type. To
this end he introduced the notion of a posit. The justification of the straight rule
with respect to the end of converging to all and only true limits does not carry over
to this different end. However, let us not throw out the baby with the bathwater.
Reichenbach taught us that we can justify induction deductively by proving it to
be the means to attaining an end that differs from usually reasoning true premises
to true conclusions.

Specifically, we can deductively justify the principle of universal induction
relative to the cognitive end of necessarily “stabilizing” on all and only adequate
universal hypotheses, i.e. hypotheses of the form ‘All Fs are G.’ For there is
a deductively valid argument which does not presuppose its conclusion, whose
premises are restricted to some definitions, and whose conclusion says the follow-
ing: in each possible world w, the principle of universal induction stabilizes on
all and only those universal hypotheses that are adequate in w. Here I identify a
possible world with an ordered set of informational inputs. The talk of adequacy,
as opposed to truth, is intended to stress this assumption. I also assume that the
information we have at any point in any possible world w is true in w as well as
“cumulative” in a sense that will be made precise momentarily.

The premises of this deductively valid argument can be restricted to some
definitions because, unlike Reichenbach’s straight rule, the principle of universal
induction does not mention numbers or anything else that requires more than first-
order logic with identity. As to the definitions, they are needed to formulate the
end which the principle of universal induction is a means to attaining, and which
is mentioned in the conclusion of this deductively valid argument. The latter says
that for all possible worlds w and all universal hypotheses h, h is adequate in w if
and only if there is a point such that at this and any later point, h may and ought
to be inferred from the information one has at these points in w by the principle
of universal induction.

Let’s formulate this conclusion more precisely with some definitions. First

23



I assume there to be a set of points P whose elements are ordered by a binary
relation � that is anti-symmetric, transitive, and total: for all s, t,u in P, if s � t
and t � s, then s = t; if s � t and t � u, then s � u; and, s � t or t � s.
Possible worlds are characterized as functions w from P into the set of literals L,
i.e. atomic sentences and their negations, over some first-order vocabulary such
that for all atomic formulas F (a): F (a) is in the range of w if and only if ¬F (a) is
not. The range of w is the set of literals ±F (a) in L for which there is a point t in
P such that w (t) =±F (a). The universal hypothesis ‘All Fs are G’ is adequate in
the possible world w : P → L if and only if there is no individual constant a such
that F (a) ∧¬G (a) follows logically from the range of w, I (w) = {w (t) : t ∈ P}.
Otherwise it is inadequate in w.

The information one has at some point t in P in a possible world w is the set
of all literals w (s) such that s � t, I (w, t) = {w (s) : s ∈ P,s � t}. This definition
implies that the information one has at any point t in any world w is true in w:
I (w, t) ⊆ I (w). It also follows that every bit of information that is true in w
will be received eventually: if some item of information belongs to I (w), then it
belongs to I (w, t), for some t in P. Finally, once received, an item of information
will never be forgotten. This is so because the information one has at any point in
any possible world is cumulative: for all points s and t, and all possible worlds w,
if s � t, then I (w,s) ⊆ I (w, t).

Let De f be a set of first-order formulas describing all these definitions. This
means that we cannot make the assumptions that P is countable and that � is
a well-ordering, because countability and well-ordering cannot be formulated in
first-order logic. As mentioned, De f is needed to formulate the end which the
principle of universal induction is a means to attaining. De f is not needed to
formulate the principle of universal induction itself. The following proof can be
carried out in first-order logic with identity.

Let w be an arbitrary possible world and h an arbitrary universal hypothesis,
i.e. statement of the form ‘All Fs are G,’ where ‘F’ and ‘G’ are arbitrary predicates
(and not just atomic ones). h is adequate in w, or h is inadequate in w. If h
is adequate in w, then there is no individual constant a such that F (a) ∧¬G (a)
follows logically from I (w). Since I (w, t) ⊆ I (w) for all points t, there is no
individual constant a such that F (a) ∧¬G (a) follows logically from I (w, t), for
any point t. Hence at any point t in w, all objects, if any, about which one has
enough information – i.e. whether they are F and whether they are G – are G
if they are F. According to the principle of universal induction, from all these
premises I (w, t) one may and ought to infer the conclusion that all Fs are G.

If h is inadequate in w, then there is an individual constant a such that F (a) ∧

24



¬G (a) follows logically from I (w). By compactness there is a finite subset I (w)−

of I (w) that logically implies F (a) ∧¬G (a). Since I (w)− is finite, there exists a
point s such that t � s for all points t with w (t) ∈ I (w)−. Since information is
cumulative it holds for all points u such that s � u: not all objects about which
one has enough information – i.e. whether they are F and whether they are G – at
point u in w are G if they are F. Therefore it holds for all points u such that s � u:
it is not the case that one may and ought to infer the conclusion that all Fs are G
from the premise I (w,u) by the principle of universal induction.

The universal hypotheses that the principle of universal induction conjectures
will, in general, be “over the top.” For instance, as long as there is no object about
which one has the information whether it is F it conjectures that all Fs are ¬F.
This boldness is necessary for conjecturing all adequate universal hypotheses “as
quickly as possible.” Before showing this let us put this first result in symbols,
and in various flavors that are not logically equivalent, but true nonetheless:

De f `DL ∀w∀h (w |= h →∃s∀t � s : I (w, t) `PoI h)
De f `DL ∀w∀h (w 6|= h →∃s∀t � s : I (w, t) 0PoI h)
De f `DL ∀w∀h (w |= h ↔∃s∀t � s : I (w, t) `PoI h)
De f `DL ∀w∀h (w 6|= h ↔∃s∀t � s : I (w, t) 0PoI h)

`DL is derivability in deductive logic DL, and `PoI is derivability from the principle
of universal induction PoI. |= is the relation of adequacy, and the variables ‘h’
and ‘w’ range over universal hypotheses and possible worlds, respectively. The
variables ‘s’ and ‘t’ range over points in P, and t � s holds if and only if s � t. The
following formula says that there is a derivation in first-order logic with identity
from our definitions to the claim that, with necessity, the principle of universal
induction stabilizes on all and only adequate hypotheses: De f `DL NecSt (PoI).

Logicians may want to think of this result in terms of correctness and com-
pleteness: the principle of universal induction stabilizes only on adequate univer-
sal hypotheses (correctness), and it stabilizes on all adequate universal hypotheses
(completeness). This means that reasoning in accordance with the principle of uni-
versal induction is a means to attaining the cognitive end of necessarily stabilizing
on all and only adequate universal hypotheses.

As in the case of the straight rule, this result does not uniquely characterize
the principle of universal induction. Uniqueness is achieved by focusing on a
more ambitious cognitive end. Among all principles that have the same abilities
and limitations as PoI (effectively, that have no computational limitations and, at
point t in possible world w, have access to, and only to, I (w, t)), the latter is such

25



that in all possible worlds w: PoI is at least as fast in w as any other principle
when it comes to inferring universal hypotheses that are adequate in w. This is so,
because PoI conjectures that all Fs are G, for any predicates ‘F’ and ‘G,’ at any
point t at which the information one has at t in any world w does not logically
imply F (a) ∧¬G (a) for some individual constant a. In particular, this is so when
one does not even have any information about any Fs or Gs – or, indeed, any
information at all. Therefore no principle is faster in any possible world when it
comes to inferring adequate universal hypotheses (as well as inadequate ones).

This more ambitious cognitive end still leaves us with several principles that
are all functionally equivalent in the sense of necessarily stabilizing on all and
only adequate universal hypotheses, and inferring adequate universal hypotheses
as quickly as possible. However, among the latter principles PoI is uniquely char-
acterized by being the one and only principle that additionally avoids inadequate
universal hypotheses as quickly as possible. For suppose there is a principle PoJ
and a possible world w and a universal hypothesis h of the form ‘All Fs are G’ that
is inadequate in w, and PoJ stops inferring h in w at some point t, before PoI does.
Consider a possible world w∗ that is identical to w in all points s with s � t, but
that differs from w insofar as h is adequate in w∗. Such a possible world w∗ exists,
because PoI still conjectures that all Fs are G at t in w, and so no I (w,s), for any
s � t, logically implies F (a)∧¬G (a) for any individual constant a. Since PoJ has
the same abilities and limitations as PoI, PoJ still – but wrongly! – stops inferring
h in this alternative possible world w∗ at point t. If PoJ never resumes inferring h
in w∗, or does so, but without stabilizing on h, then PoJ does not stabilize on all
universal hypotheses that are adequate in w∗. If PoJ resumes inferring h in w∗ and
stabilizes on h, then it does not do so as fast as PoI. In either case PoJ is not one
of those principles that necessarily stabilize on all and only adequate universal
hypotheses, and infer adequate universal hypotheses as quickly as possible.

Thus the principle of universal induction is the one and only means to attaining
the cognitive end of necessarily stabilizing on all and only adequate universal
hypotheses, of inferring adequate universal hypotheses as fast as possible, and of
subsequently avoiding inadequate universal hypotheses as fast as possible. (I write
‘subsequently’ to stress that the former end has priority over the latter.)

The work by Reichenbach (1938; 1940) and others (Putnam 1963, Gold 1967)
has inspired an entire discipline of deductive justifications of induction: formal
learning theory. It is developed as brilliantly as it is presented beautifully in Kelly
(1996) and Martin & Osherson (1998). For the purposes of this paper the variant
of Reichenbach’s deductive justification of the principle of universal induction just
given is all we need. Let us now turn to the inductive justification of deduction.

26



7 Haack’s Argument against Deduction
Carnap (1968) and Goodman (1954) compare induction to deduction. Haack
(1976) compares deduction to induction. Her preferred variant of Hume (1739;
1748)’s argument against induction takes the form of a dilemma.

Hume’s dilemma for induction A deductive justification of induc-
tion would be too strong, because it would show that induction always
leads from true premises to true conclusions.

An inductive justification of induction would be circular.

Haack’s dilemma for deduction An inductive justification of deduc-
tion would be too weak, because it would merely show that deduction
usually leads from true premises to true conclusions.

A deductive justification of deduction would be circular.

Haack (1976) is right in claiming that an inductive justification of induction would
be circular, and that a deductive justification of deduction would be circular. These
are two immediate circles that I will return to below. However, even if we grant
Premise 1 of Hume’s Argument, Haack (1976) is not right in claiming that a de-
ductive justification of induction would be too strong. It is true, as we have seen,
that there is no deductively valid argument which does not presuppose its conclu-
sion, whose premises are restricted to information we have, and whose conclusion
says that the principle of induction usually leads from true premises to true con-
clusions. And perhaps this is what Haack (1976) means. It is not true, though,
that the latter conclusion would be too strong to justify the principle of induction.

It is also not true that an inductive justification of deduction would be too
weak, or so I will argue now. We have the following piece of information:

All particular inferences which are such that we have the information
(i) whether their ingredient sentences (i.e. premises plus conclusion)
are true or false, and (ii) whether they conform to the general rules
of classical deductive logic, have at least one false premise or a true
conclusion if they conform to the general rules of deductive logic.

From this premise we get the following conclusion by an application of the prin-
ciple of universal induction:

All particular inferences that conform to the general rules of classical
deductive logic have at least one false premise or a true conclusion.

27



This conclusion says that classical deductive logic always leads from true premises
to true conclusions. It does not merely say that classical deductive logic usually
leads from true premises to true conclusions. Since the argument also does not
presuppose its conclusion, and its premise is restricted to information we have,
we have a perfectly fine inductive justification of deduction. Pace Haack (1976) it
is not the case that an inductive justification of deduction would be too weak.

More precisely, we have an inductive justification of classical deductive logic
relative to the cognitive end of reasoning in an actually truth-preserving way.
What we do not have is an inductive justification of classical deductive logic rel-
ative to the different cognitive end of reasoning in a way that is truth-preserving
with logical necessity: the conclusion of our inductive justification of classical
deductive logic does not say that the particular inferences conforming to the gen-
eral rules of classical deductive logic lead from true premises to true conclusions
with logical necessity. And perhaps this is what Haack (1976) means when she
says that an inductive justification of deduction would be too weak. The con-
clusion merely says that particular inferences which conform to the general rules
of classical deductive logic actually and always lead from true premises to true
conclusions. However, the conclusion does not merely say that particular infer-
ences which conform to the general rules of classical deductive logic normally or
usually lead from true premises to true conclusions.

Even if this were the complaint, it is not the principle of induction that is
to be blamed. If anyone, it is we who are to be blamed, because our cognitive
limitations prevent us from establishing the premise needed to inductively infer
this stronger conclusion without presupposing it. Suppose we had the following
information:

All particular inferences which are such that we have the information
(i) whether their ingredient sentences (i.e. premises plus conclusion)
are true or false, and (ii) whether they conform to the general rules of
classical deductive logic, are such that, if they conform to the general
rules of deductive logic, then, with logical necessity, at least one of
their premises is false or their conclusion is true.

From this premise we would indeed get the following stronger conclusion by an
application of the principle of universal induction:

All particular inferences that conform to the general rules of classical
deductive logic are such that, with logical necessity, at least one of
their premises is false or their conclusion is true.

28



The problem is that we cannot show that the particular inferences which are
such that we have the information whether their ingredient sentences are true or
false, and that conform to the general rules of classical deductive logic, are truth-
preserving with logical necessity without presupposing that particular inferences
that conform to the general rules of classical deductive logic are truth-preserving
with logical necessity. The problem is not the principle of universal induction,
which would happily yield this conclusion for us if our cognitive limitations did
not prevent us from having as information the premise needed to inductively infer
this stronger conclusion.

Besides being able to justify deduction inductively relative to the cognitive
end of reasoning in an actually truth-preserving way, we saw in section 6 that
we can also justify various principles of induction deductively relative to various
cognitive ends (and, of course, inductively, as many philosophers have noted).
Moreover, we can justify deduction deductively relative to the end of reasoning
in a way that is truth-preserving with logical necessity, as we have seen in section
5. To be sure, this deductive justification of deduction is circular. However, it is
neither trivial nor need this circularity concern us.

It is not trivial, because the rules of classical deductive logic do not justify
themselves deductively relative to all ends (for instance, they do not justify them-
selves relative to the end of reasoning in a normally truth-preserving way). Nor
do they justify the rules of other systems of logic (for instance, the non-monotonic
logic by Kraus & Lehmann & Magidor 1990) with respect to the end of reason-
ing in a way that is truth-preserving with logical necessity. Nor do they justify
themselves relative to the end of reasoning in a way that is truth-preserving with
logical necessity once we enrich them to include further axioms.

The circularity need not concern us, because classical deductive logic can be
justified inductively relative to the end of reasoning in an actually truth-preserving
way. While other systems of logic can also justify themselves in a circular way
relative to some end, none of these other systems of logic will be justified induc-
tively relative to the end of reasoning in an actually truth-preserving way. This is
so because of our previously established result that all general rules of classical
deductive logic are actually truth-preserving and, as we will see now, because only
the general rules of classical deductive logic are actually truth-preserving.

The latter is the case for the following reason: any system of logic that contains
at least one general rule that is not already part of classical deductive logic will be
such that there is at least one particular inference that conforms to this general rule
and has true premises, but a false conclusion. (Logicians call rules that go beyond
classical deductive logic inadmissible in it. They call the rules schemata. And

29



they point out that the rules or schemata must not mention any logical constants
other than those of classical first-order logic with identity.) Call a general rule of
logic tested by us if and only if there is at least one particular inference of which
we have have the information whether its ingredient sentences are true or false,
and that it conforms to this rule. I assume we have the following information:

For any general rule of logic that we have tested, and of which we
have the information whether it goes beyond classical deductive logic,
if it goes beyond classical logic, then there is at least one particular
inference that conforms to this rule, whose premises are true, and
whose conclusion is false.

What I am assuming here is that we can provide a “counterexample” to any rule
that we have tested and of which we have the information that it is inadmissible
in classical logic. That is, we can provide a particular inference of which we have
the information that it conforms to the rule, that its premises are true, and that its
conclusion is false. An application of the principle of universal induction to the
above piece of information gives us the following conclusion:

For all general rules of logic that go beyond classical deductive logic
there is at least one particular inference that conforms to this rule,
whose premises are true, and whose conclusion is false.

Logicians may want to think of this conclusion as a completeness result (only the
general rules of classical deductive logic are actually truth-preserving), and of the
previous one as a correctness result (all general rules of classical deductive logic
are actually truth-preserving).

Consequently we have an inductive justification of the general rules of classi-
cal deductive logic relative to the cognitive end of reasoning in an actually truth-
preserving way: there is an inductively strong argument which does not presup-
pose its conclusion, whose premises are restricted to information we have, and
whose conclusion says that all and only the general rules of classical deductive
logic are actually truth-preserving. Here a general rule of logic is actually truth-
preserving just in case all the particular inferences conforming to it have at least
one false premise or a true conclusion. In symbols (and combining ∀x (Ax → Bx)
and ∀x (Bx → Ax) to ∀x (Ax ↔ Bx) for the sake of perspicuity):

I (@,now) `PoI ∀R (TP@ (R) ↔ AdmDL (R))

I (@,now) `PoI ∀R (∀iR : @ |= iR ↔ ∀iR : `DL iR)

30



TP@ is actual truth-preservation, and AdmDL is admissibility in deductive logic
DL. |= now is the relation of truth, and I (@,now) is the information we have now
in the actual world. The variables ‘R’ and ‘iR’ range over rules and instances,
respectively. An instance is identified with a disjunction whose disjuncts are the
negations of all premises, and the conclusion. The following formula says that
there is an application of the principle of universal induction whose premise is the
information we have now in the actual world and whose conclusion says that first-
order logic with identity is actually truth-preserving: I (@,now) `PoI ActTP (DL).

The inductive justification of classical deductive logic obtains relative to the
end of reasoning in an actually truth-preserving way. The deductive justifica-
tion of the principle of universal induction obtains relative to the different end of
necessarily stabilizing on all and only adequate universal hypotheses (and, if one
chooses to add so, of inferring adequate universal hypotheses as quickly as possi-
ble, and of subsequently avoiding inadequate universal hypotheses as quickly as
possible). One might think I have replaced two immediate circles, the inductive
justification of induction and the deductive justification of deduction, by one not
so immediate, but still circular circle. This would be wrong, though.

It is not the case that we justify, in some logic L, deduction inductively, and
induction deductively, let alone that we justify the package consisting of both in-
duction and deduction in a deductive-or-inductive – or any other – way. Nor is it
the case that we justify, in some logic L, the claim that deductive logic is (actually
or necessarily) truth-preserving by assuming inductive logic to be (actually or nec-
essarily) stabilizing on all and only adequate universal hypotheses, and the claim
that inductive logic (actually or necessarily) stabilizes on all and only adequate
universal hypotheses by assuming deductive logic to be (actually or necessarily)
truth-preserving. That is, neither of the following is the case:

PoI `L DL `L PoI or DL `L PoI `L DL

DL&PoI `L DL&PoI or X `DL&PoI DL&PoI or X `L PoI&DL

ActSt (PoI) `L ActTP (DL) `L ActSt (PoI)

ActSt (PoI) `L NecTP (DL) `L ActSt (PoI)

NecSt (PoI) `L ActTP (DL) `L NecSt (PoI)

NecSt (PoI) `L NecTP (DL) `L NecSt (PoI)

Rather, we use induction to work our way towards an inductive justification of
deduction relative to one cognitive end. And we use deduction to work our way

31



towards a deductive justification of induction relative to a different cognitive end.
As long as these two cognitive ends are different, we are not caught in a circle.

I (@,now) `PoI ActTP (DL) and De f `DL NecSt (PoI)

Maybe it helps to illustrate the situation with an analogy. Let me begin with a
disanalogy. Ingrid and Desmond are two friends of mine. Ingrid tells me that all
and only those claims are true that Desmond claims to be true. Desmond tells me
that all and only those claims are true that Ingrid claims to be true. Suppose what
Desmond tells me is true: all and only those claims are true that Ingrid claims to
be true. Then Ingrid’s claim is true that all and only those claims are true that
Desmond claims to be true. And so Desmond’s claim is true that all and only
those claims are true that Ingrid claims to be true. Now suppose what Desmond
tells me is false. Then Ingrid’s claim is false that all and only those claims are true
that Desmond claims to be true. And so Desmond’s claim is false that all and only
those claims are true that Ingrid claims to be true. If this were our situation, we
would indeed be caught in a circle.

However, this is not our situation. Our situation is more analogous to the
following one. Ingrid and Desmond are two friends of mine. Their expertise lies
in different, though overlapping areas, and while they speak the same language,
the way they speak is very different. Ingrid refines her claims over time, and while
her claims are restricted to the location of their utterance, they are most interesting.
Desmond, on the other hand, only makes seemingly platitudinous claims and he
has already said everything he had to say. While Desmond’s claims were not very
bold, they were claimed to be true everywhere. Ingrid’s claims vary with time and
location, Desmond’s do not.

Ingrid has just reassured me (as she has always as far as I remember) that the
following is true here: all and only those general rules are truth-preserving here
that Desmond claimed to be truth-preserving everywhere.

Desmond has claimed that the following is true everywhere: wherever Ingrid
has been, or will be, speaking, all and only those universal hypotheses are true
there that Ingrid there eventually continues to claim to be true there forever.

In the first case Ingrid and Desmond praise each other of the same qualities in
the same way. In this sense the first case is circular. In the second case Ingrid and
Desmond also praise each other, but they praise each other of different qualities,
and they praise each other in very different ways. In the first case Ingrid’s claim
is true just in case Desmond’s claim is true, which just means that Ingrid’s claim
is false just in case Desmond’s claim is false. This is circularity, but it is absent in
the second case, which illustrates our situation.

32



Appendix
The following sentence schema S gives rise to a possible objection to the inductive
completeness argument:

∃x¬∃yP
(
y,x
)
∧∀x∃yP

(
x, y
)
∧∀x∀y∀z

(
P (x,z) ∧ P

(
y,z
)
→ x = y

)
No instance of S is true in the model-theoretic sense in any finite model. However,
the instance of S that replaces ‘P’ with ‘the natural number ... is a precursor of the
natural number ...’ is true. That is, S is true in the model whose domain is the set of
natural numbers and whose interpretation function interprets ‘P’ as the precursor
relation on the natural numbers. Therefore every instance of the negation of S,
¬S, is true in every finite model, but some instance of ¬S is false in some infinite
model. This means that the general rule of logic ∴¬S is not valid.

Here is the objection. Presumably the proponent of the inductive completeness
argument is an empiricist who holds that every particular inference of which she,
i.e. the empiricist, has the information that it conforms to this general rule, and
whether its ingredient sentences are true or false, has (at least one false premise
or) a true conclusion. After all, how could an empiricist claim something that is
only true in infinite models. Yet if this is so, then this general rule comes out as
valid on the inductive completeness argument, even though it is not!

My response to this objection is that it applies a double standard. Either the
objector has the information that the instance of S described above (or some other
instance) is true, or else she does not. If, as her objection suggests, she does, then
she, i.e. the objector, does not have the information that every particular inference
of which she, i.e. the objector, has the information that it conforms to this general
rule, and whether its ingredient sentences are true or false, has (at least one false
premise or) a true conclusion. In this case the above rule does not come out as
valid on the inductive completeness argument. If, on the other hand, the objector
does not have the information that the instance of S described above (or some
other instance) is true, then she does not have the information that the rule is not
valid. In this case she cannot raise her objection.

Thus this objection to the inductive completeness argument rests on applying
a double standard. The objector assumes to have information about arithmetical
claims that she does not allow herself to possess when carrying out the induc-
tive completeness argument. Put differently, the objection denies the empiricist
proponent of the inductive completeness argument information about arithmeti-
cal claims that she must suppose her to possess if the objection is to get off the
ground.

33



Acknowledgments
I am grateful to, among others, Jake Chandler, Richard Dawid, Rory Harder,
Ole Hjortland, David Hyder, Norbert Gratzl, Terence Irwin, Hannes Leitgeb,
Paul Rusnock, Gregory Wheeler, Timothy Williamson, and, especially, Wolfgang
Spohn and Marta Sznajder for many helpful comments and suggestions on earlier
versions of this paper.

My research was supported by the Canadian SSHRC through its Insight pro-
gram and by the Connaught Foundation through its New Researcher program.

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38


	Introduction
	Hume's Argument against Induction
	Carnap's Rationalist View of Rationality
	The Continuum of Carnap's Rationalism
	Goodman's Fiction about the Facts of Forecast
	Reichenbach Rules
	Haack's Argument against Deduction