1. Introduction

Enumerative induction says that a law is confirmed by its positive instances or may be inductively inferred from them (in the absence of negative instances). It is, for sure, the most venerable and primitive of all inductive rules. But, it has a bad press. It is very crude; science does not seem to proceed with such simple rules. Goodman's new riddle of induction has shown that enumerative induction is inconsistent, if generally applied; it seems impossible to say what the appropriate restrictions are. On the face of it, it is a rule of qualitative confirmation theory; but philosophers have despaired of constructing such a theory.

The rule has finally found a Bayesian home. It is true, though, that at least within inductive logic as developed by Carnap (Reference Carnap, Carnap and Jeffrey1971, Reference Carnap and Jeffrey1980), nothing can confirm a law because each law has probability 0 (if its domain of quantification is infinite). The natural idea was then to turn enumerative induction into the principle of positive instantial relevance, according to which each positive instance confirms that the next instance is also positive. This seems reasonable, and accepted. So, why bother any longer?

Well, ‘primitive’ is ambiguous. It may, indeed, mean ‘not workable’. But, it also means ‘basic’. If we do not fully understand the basic things, how can we ever hope to come to terms with the more complicated things? So whoever is concerned with inductive, plausible, or uncertain reasoning should be concerned with understanding such a primitive rule as enumerative induction. The goal of the paper is to enhance this understanding. The way to reach the goal is to bring enumerative induction home from quantitative to qualitative confirmation theory, and the reason why this is feasible is that in the meantime, we have a fully general qualitative confirmation theory at our disposal, namely, ranking theory. This needs a little explanation.

Traditionally, confirmation theory is a field within philosophy of science. Its quantitative or probabilistic version, i.e., Bayesianism, has been a major option from the beginning. In the ’50s and ’60s we also saw forceful attempts to construct a qualitative confirmation theory. However, the project was abandoned in the ’70s, for reasons nicely summarized in Niiniluoto (Reference Niiniluoto1972). Thus, at least within philosophy of science, Bayesianism had won the day. However, logicians and computer scientists were very active since around 1975 in producing alternatives, though rarely under the labels ‘induction’ or ‘confirmation’ (see, e.g., the many theories collected in Gabbay and Smets Reference Gabbay and Smets1998–2000, in particular vols. 1 and 3), and hence scarcely noticed in epistemology and philosophy of science. The multiplicity of proposals developed there is quite confusing. Still, I believe that ranking theory as developed by me (Spohn Reference Spohn1983, sec. 5.3; Reference Spohn, Harper and Skyrms1988), though under a different name, is the most suitable qualitative account of induction or confirmation.

This introduction is not the place for extensively arguing the case; the old reasons given in Spohn Reference Spohn, Harper and Skyrms1988 still apply. Let me state only the most important point: The central notion in this connection is the notion of conditional belief. In order to say whether some evidence would qualitatively confirm a given hypothesis, we must, to put it vaguely, consider whether the hypothesis would be believed given the evidence and whether it would be believed without the evidence. If we want to give an account of induction, we have to give an account of belief change, which I have argued in Spohn Reference Spohn and Enskat2000. And, belief change best works by conditionalization rules that essentially refer to conditional beliefs, just as probabilistic conditionalization rules refer to conditional probabilities. We do need an adequate notion of conditional belief.

Hence, we should look at the various attempts to explain it. Belief revision theory (cf. e.g., Gärdenfors Reference Gärdenfors1988) makes a plausible proposal: B is believed, given A in a certain belief state, iff B is believed in the revision of that state by A. But, as I have argued in Spohn Reference Spohn, Harper and Skyrms1988, belief revision theory, as it is presented up to date, is defective and the proposal is therefore inadequate. One might say that B is believed, given A, iff $P(B\vert A) =1$ , but this proposal is incomplete, because in standard probability theory, this conditional probability is undefined if $P(A) =0$ . One might insist on the proposal by interpreting P as a Popper measure that fills this incompleteness by taking conditional probability as an undefined primitive. However, as shown in Spohn Reference Spohn1986, this idea is defective in just the way belief revision theory is, and so on. In the end, I claim, one must turn to ranking theory that offers the most adequate account of conditional belief.

Here, I simply want to proceed on the basis of this scarcely redeemed claim. The point of the introductory remarks was only to suggest that the most promising way to study enumerative induction is in terms of ranking functions. This is what I want to do here. Hence, the plan of the paper is this: In Section 2, I shall introduce the theory of ranking functions as far as we need it here. Section 3 will then apply ranking theory to enumerative induction, which as we shall see, may be instantiated by a variety of schemes. This will turn out to be a brief and rather boring exercise; the insights come later. In Section 4, I shall propose a ranking-theoretical explication of what a possible law, or a nomological hypothesis, is. In Section 5, we shall be able to show that there is a one to one correspondence between schemes of enumerative induction as found in Section 3 and mixtures of nomological hypotheses as explained in Section 4. Thus, our ranking theoretic analysis will result in transferring de Finetti's deep account of the confirmation of statistical hypotheses to the deterministic or qualitative realm. Section 6 will conclude with some remarks on the defeasible or unrevisable apriority of lawfulness or the uniformity of nature.

2. Ranking Functions

Let us start with a set W of possible worlds, small worlds in the sense of Savage, rather than maximally large worlds in the sense of Lewis. Each subset of W is a truth condition or proposition. Hence, the set of propositions forms a complete Boolean algebra. I shall assume outright propositions to be the objects of doxastic attitudes, thereby taking these attitudes to be intensional. We know well that this is problematic—that the so-called propositional attitudes are presumably hyperintensional. But, we scarcely know what to do about the problem. Hence, my assumption is just to signal that I do not want to worry here about these kinds of problems.Footnote ¹

Moreover, I assume that there is a distinguished class of (logically independent) atomic propositions. The paradigmatic atomic proposition states that a certain object has a certain property. Finally, I shall assume that the complete algebra of propositions is generated by the atomic propositions. Thus, each possible world is tantamount to a maximally consistent, and possibly infinite, conjunction of atomic propositions. A proposition is called molecular iff it is a member of the Boolean algebra generated by the atomic propositions, i.e., iff it is generated from the atomic propositions by finitely many Boolean operations.Footnote ²

This is all we need to introduce our basic notion:

Since singletons of worlds are propositions as well, the point and the set function are interdefinable. The point function is simpler, but auxiliary, the set function is the one to be interpreted as a doxastic state.

Indeed, ranks are best interpreted as degrees of disbelief. $\kappa (A) =0$ says that A is not disbelieved at all; $\kappa (A) =1$ says that A is disbelieved (and hence $\overline{A}$ believed) to degree 1, etc. Note that $\kappa (A) =0$ does not say that A is believed; this is rather expressed by $\kappa (\overline{A}) > 0$ , i.e., that non-A is disbelieved (to some degree). The clause that $\kappa (w) =0$ for some $w\in W$ is thus a consistency requirement. It guarantees that at least some proposition, and in particular W itself, is not disbelieved (and hence that some proposition, e.g., Ø, is not believed). This entails

The set $C_{\kappa }=\{ w\vert \kappa (w) =0\} $ is called the core of κ (or of the doxastic state represented by κ). C _κ is the strongest proposition believed (to be true) in κ. In fact, a proposition is believed in κ if and only if it is a superset of C _κ. Hence, the set of beliefs is deductively closed according to this representation.Footnote ⁴

These observations make clear the following essential point: On the one hand, the degrees of disbelief are the basic notion. On the other hand, these degrees also contain an all-or-nothing notion of disbelief (and thus belief): disbelief is disbelief to some positive degree. If we would confine ourselves to a static perspective, this all-or-nothing notion, which I sometimes called plain (dis-)belief and which is well studied in doxastic logic, would be good enough. However, in order to define an adequate notion of conditional belief, and thus to account for the dynamics of the all-or-nothing notion, we have to introduce the degrees. I emphasize this point because it marks an important advantage of ranking over probability theory. The latter cannot offer an adequate notion of plain belief, and hence those raised in probabilistic thinking tend to find the notion disreputable. But intuitively we have the notion, and it is basic to large parts of epistemology. Ranking theory satisfies both needs here, the one for the all-or-nothing notion and the other for the degrees.

There are two laws for the distribution of degrees of disbelief. One is

That is, the degree of disbelief in A and the degree of disbelief in B given A add up to the degree of disbelief in A-and-B; this appears highly intuitive. With Definition 1, we may say, conversely, that this is precisely how conditional degrees of disbelief are to be understood. And the other is

That is, the degree of disbelief in a disjunction is the minimum of the degrees of the disjuncts. Given the definition of conditional ranks, this law is nothing but a conditional consistency requirement; if it would not hold the inconsistency could arise that both $\kappa (A\vert A\cup B) $ and $\kappa (B\vert A\cup B) > 0$ , i.e., that both A and B are disbelieved, given A-or-B.

According to Definition 1, the law of disjunction indeed extends to disjunctions of arbitrary cardinality. I find this reasonable, since an inconsistency is to be avoided in any case, be it finitely or infinitely generated. Note that this entails that each countable set of ranks must have a minimum (not only an infimum) and that the range of a ranking function must, therefore, be well ordered. Hence, the range N⁺ is a natural choice. This point will become important later on.Footnote ⁵

I immediately add

In the following, we shall consider only regular ranking functions. In earlier papers, I have assumed a stronger form of regularity by outright defining a ranking function to be a function from W into N so that only Ø receives infinite rank. If all propositions are molecular, there is no difference. In this paper, however, we want to consider possibly infinite, and thus nonmolecular, generalizations, and then this stronger form of regularity is not feasible. Hence the present weaker assumption.

There is no need here to develop ranking theory more extensively. A general remark may be more helpful: Ranking theory works in almost perfect parallel to probability theory. Take any probabilistic theorem, replace probabilities by ranks, the sum of probabilities by the minimum of ranks, the product of probabilities by the sum of ranks, and the quotient of probabilities by the difference of ranks, and you are almost guaranteed to arrive at a ranking theorem. Additivity of probabilities thus translates into the law of disjunction for ranks. The probabilistic law of multiplication translates into the above law of conjunction. It is easy to prove the ranking analogue to the formula of total probability, the

which says, for a partition {B ₁, …, B _n} of W, how to compute the rank of some proposition A from the rank of A given various hypotheses B _i and the ranks of the hypotheses B _i themselves. One may continue with a ranking version of Bayes’ theorem.Footnote ⁶ One can even develop the whole theory of Bayesian nets in ranking terms,Footnote ⁷ and so on.

The general reason is that ranks may roughly be interpreted as orders of magnitude of (infinitesimal) probabilities. Consider a nonstandard probability measure taking nonstandard reals as values. The logarithm of a product of such probabilities is the sum of the logarithms of the factors, wrt any base. And, the order of magnitude (the logarithm in round figures) of a sum of such probabilities is the minimum of the orders of magnitude of its terms, at least wrt an infinitesimal base. This perspective explains the translatability. However, I should emphasize that the translation is only an excellent rule of thumb, but not perfectly reliable, as we shall see later on (cf. also Spohn Reference Spohn and Humphreys1994). The matter is not fully cleared up.

It is still annoying, perhaps, that belief is not characterized in a positive way. But there is a remedy.

Recall that at least one of the terms $\kappa (\overline{A}) $ and κ(A) must be 0. Hence, $\beta (A) > 0$ , < 0, or = 0 iff, respectively, A is believed, disbelieved, or neither; and the larger β(A), the more strongly A is believed. Thus, belief functions may appear to be more natural. But, their formal behavior is more awkward. I shall use both notions.

Since this is an essay about confirmation theory, we must ask: What is confirmation with respect to ranking functions? The same as elsewhere, namely, positive relevance.

There is an issue here whether the condition should require $\beta (B\vert A) > \beta (B) $ or only $\beta (B\vert A) > \beta (B\vert \overline{A}) $ , as stated. In the corresponding probabilistic case, the two conditions are equivalent if all three terms are defined, but the first condition is a bit more general, since it may be defined, while the second is not. That is why the first is often preferred. In the ranking case, however, all three terms are always defined, and the second condition may be satisfied, while the first is not. In that case, the second condition on which my definition is based seems to be more adequate.Footnote ⁹

Let me close my presentation of ranking theory with formally introducing a point that will receive great importance later on: Ranking functions can be mixed, just as probability measures can. For instance, if κ₁ and κ₂ are two ranking functions for W and if κ* is defined by

then κ* is again a ranking function for W. Or, more generally:

Definition 5. Let K be a set of ranking functions for W and ρ a ranking function for K. Then κ* defined by

is (obviously) a ranking function for W and is called the mixture of K by ρ.

Note the similarity of this definition with the formula of the total rank; the various κ take here the role of the various hypotheses B _i in that formula.

3. Symmetry and Nonnegative Instantial Relevance

Now we are well prepared to turn to our proper topic: enumerative induction. Let us start with simplifying the propositional structure as far as our topic allows, by considering an infinite series of objects and just one property P. So, each object can either have or lack P, and there are just two universal generalizations: ‘all objects are P', and ‘all objects are not P’. Concerning the objects, this is all the generality we need; concerning the properties we proceed minimally. This will facilitate our business. It will be clear, though tedious to prove, that the results below generalize to any finite number of properties. So, the results are considerably stronger than they appear. However, I don't know how things stand with an infinity of properties that may be generated, e.g., by a real-valued magnitude.

This simplification allows us to represent each possible world by a sequence $\boldsymbol{z}=(z_{1},\,z_{2},\,\ldots) $ of 1's and 0's, where $z_{n}=1$ or 0 means, respectively, that the n-th object has or lacks P. {x takes $z_{i_{1}},\,\ldots,\,z_{i_{n}}$ } is short for the proposition $\{ \boldsymbol{x}\vert x_{i_{j}}=z_{i_{j}}\,\mathrm{for}\,\,j=1,\,\ldots,\,n\} $ .

The most basic assumption ranking functions will be supposed to satisfy is symmetry. This means that ranking functions should be able to distinguish different objects only with respect to the properties considered, in our case P and non-P. Let us make this precise in the following definition:

Regular symmetric ranking functions take a particularly simple form, as stated in the obvious

Theorem 1. For each regular symmetric κ there is a representative function f from $\mathbf{N}\times \mathbf{N}$ into N such that $\kappa (\boldsymbol{x}\,\mathrm{takes}\,\,z_{1},\,\ldots,\,z_{m+n}) =f(m,\,n) $ if $\sum_{i=1}^{m+n}z_{i}=m$ , i.e., if exactly m of the first $m+n$ objects have P and the others lack P. This function satisfies $f(0,\,0) =0$ and the minimum property $f(m,\,n) =\mathrm{min}\,[ f(m+1,\,n),\,f(m,\,n+1) ] $ (for a proof apply the law of disjunction to the fact that

Conversely, any function f from $\mathbf{N}\times \mathbf{N}$ into N with these two properties represents a regular symmetric ranking function.

This entails that f can be visualized as an infinite triangle of nonnegative integers

If a path in such a triangle is any sequence which starts at any point $f(m,\,n) $ and in which each member is succeeded by its left or right neighbor immediately below, then the minimum property entails that each such path is non-decreasing and that whenever a path increases by going left, any path going right at this point does not increase, and vice versa.

Symmetry has a long and venerable history. Indeed, van Fraassen (Reference van Fraassen1989) even went so far as to argue that lawlikeness is a confused idea we should dispense with, and that symmetry takes the key role in scientific reasoning in its place. This paper will, in fact, confirm van Fraassen's view, with the minor divergence being that lawlikeness need not be dispensed with, but instead will receive an appropriate account through the notion of symmetry. In any case, we shall pursue our investigation of enumerative induction only in terms of symmetric (and regular) ranking functions.

The first noteworthy observation in this pursuit is that given symmetry, there is no difference between belief in the next instance and belief in the universal generalization about all further instances. Suppose that after some evidence concerning the first n objects you believe that the $n+1$ object will have P; that is, your κ is such that $\kappa (\boldsymbol{x}\,\mathrm{takes}\,\,z_{n+1}=0) =s$ for some $s> 0$ . Because of symmetry you then believe that any further object will have P, that is, $\kappa (\boldsymbol{x}\,\mathrm{takes}\,\,z_{n+k}=0) =s> 0$ for any $k\geq 1$ . And because of the infinite variant of the law of disjunction, this entails that you believe in all further objects having P with the same strength; that is, your disbelief that some future object lacks P is as strong as your disbelief that a specific future object lacks P; i.e., $\kappa (\bigcup _{k\geq 1}\boldsymbol{x}\,\mathrm{takes}\,\,z_{n+k}=0) =s> 0$ . If this sounds counter-intuitiveFootnote ¹⁰, we have to return arguing about the law of disjunction and the conditional consistency it reflects. However, don't be confused; your disbelief that all further objects lack P may still be much stronger, or even infinite.

This means that as far as positive confirmation is concerned, i.e., confirmation that generates or strengthens belief instead of merely diminishing disbelief, there is no difference between the next or any other positive instance and the universal generalization about all further instances. Hence, Carnap's problem of the null confirmation of universal generalizations disappears in the ranking theoretic context, and the recourse to instantial relevance, which was only a substitute in the Bayesian framework, is fully legitimate here.Footnote ¹¹

Instantial relevance can take a stronger and a weaker form. The principle of positive instantial relevance (PIR) says that, given any evidence concerning the first n objects, the $n+1$ object having or lacking P confirms, respectively, the $n+2$ object having or lacking P. The weaker principle of nonnegative instantial relevance (NNIR) requires only that the contrary is not confirmed. Hence, let us state

Definition 7. A regular symmetric ranking function κ satisfies PIR iff

whenever $z_{n+1}=z_{n+2}$ , i.e., iff for the relevant representative function f and all m, $n\in \mathbf{N}$

κ satisfies NNIR iff the weak inequalities hold instead.

PIR may look like the correct formalization of enumerative induction; alas, we have

Proof. Let us try to satisfy PIR by an appropriate representative function f. So we start with $f(0,\,0) =0$ and without loss of generality, $f(1,\,0) =0$ and $f(0,\,1) =r\geq 0$ . This entails $f(2,\,0) =0$ . Hence, if we set $f(1,\,1) =r$ , we already violate PIR. So, we must choose $f(1,\,1) =s> r$ and $f(0,\,2) =r$ . This in turn entails $f(3,\,0) =0$ and $f(0,\,3) =r.$ But we cannot complete, then, the fourth line of our triangle: We must set $f(2,\,1) $ or $f(1,\,2) =s$, and both choices violate PIR. □

This failure should not come as a surprise. If we try to satisfy PIR with respect to the positive instances and increase the disbelief in a negative instance with increasing positive evidence, we cannot at the same time satisfy PIR with respect to the negative instances. For, many negative instances are then just as disbelieved as a single one, and hence the negative instances cannot be positively relevant to further negative instances. We cannot have it both ways.

Hence, we are forced to settle for the weaker NNIR. It is easily seen to be consistent. Within a probabilistic setting nonnegative instantial relevance is in fact entailed by symmetry (cf. Humburg Reference Humburg, Carnap and Jeffrey1971). Thus it is worth noting that this is not the case here; it is obvious that there are symmetric ranking functions violating NNIR (because there are representative functions violating the additional condition of Definition 7).

Where do we stand? If we want to account for enumerative induction within the ranking theoretic setting, we have to accept the second best explication, i.e., NNIR. We should also keep in mind that, within this setting, positively confirming the next instance is tantamount to confirming the corresponding generalization. Thus, we may preliminarily conclude that each symmetric ranking function satisfying NNIR is a way to realize enumerative induction, there being indeed an infinity of such ways.

Still, the preliminary conclusion does not look right. There is a definite loss in the retreat from PIR to NNIR. Even partial instantial irrelevance does not really seem compatible with enumerative induction; it is strange that the confirming effect of a positive instance must fail at least sometimes. This is a negative illusion, though. In Section 5 all doubts dissolve. We shall find that NNIR is exactly right and that, contrary to appearance, positive relevance can be fully reestablished.

Our investigation has remained superficial so far. The topic gains depth only when we remind ourselves of the fact that enumerative induction was never taken to apply to all universal generalizations whatsoever, but rather only to laws or potential laws; at most with respect to laws it may claim to be a reasonable rule of inductive inference. Where is this crucial point reflected in our ranking theoretic explication? Well, it is reflected, but not at all in an obvious way. In order to uncover it, we have to think about what lawlikeness may mean in ranking theoretic terms.

4. Laws

In our simple setting we considered just two universal generalizations: $G_{1}=(1,\,1,\,\ldots) $ and $G_{0}=(0,\,0,\,\ldots) $ . What could it mean to treat G ₁, say, as a law and not as an accidental generalization? I think, quite unoriginally, that this shows in our inductive behavior. To believe in G ₁ as a law is, first, to believe in G ₁, as expressed by $\kappa (\overline{G}_{1}) > 0$ . But, as we already know, the belief in G ₁ can be realized in many different ways; this belief alone does not fix the inductive relations between the various instances. Which forms may they take? Well, if you learn about positive instances of G ₁, you do not change your beliefs about the further instances according to κ, since you expected them to be positive, anyway. Crucial differences emerge only when we look at how you respond to negative instances according to the various attitudes. Let me focus for a while on two particular responses, which I call the ‘persistent’ and the ‘shaky’ attitude.

If you have the persistent attitude, your belief in further positive instances is unaffected by negative ones, i.e., $\kappa (\boldsymbol{x}\,\mathrm{takes}\,\,z_{n+1}=0) =\kappa (\boldsymbol{x}\,\mathrm{takes}\,\,z_{n+1}=0\vert \boldsymbol{x}\,\mathrm{takes}\,\,z_{1}=\cdots =z_{n}=0) $ . If, by contrast, you have the shaky attitude, your belief in further positive instances is destroyed by a negative instance, i.e., $\kappa (\boldsymbol{x}\,\mathrm{takes}\,\,z_{2}=0\vert \boldsymbol{x}\,\mathrm{takes}\,\,z_{1}=0) =0$ , and, due to symmetry, also by several negative instances.

The difference is, I find, characteristic of the distinction between lawlike and accidental generalizations. Let us look at two famous examples. First the coins:

1. 1. All Euro coins are round.

2. 2. All of the coins in my pocket today are made of silver.

It seems intuitively clear to me that we have the persistent attitude towards 1 and the shaky one towards 2. If we come across a cornered Euro coin, we wonder what might have happened to it, but our confidence that the next coin will be round again is not shattered. If, however, I find a copper coin in my pocket, my expectations concerning the further coins simply collapse; if 2 has proved wrong in one case, it may prove wrong in any case.

Or, look at the metal cubes, which are often thought to be the toughest example, because they display no perspicuous syntactic or semantic difference:

1. 3. All solid uranium cubes are smaller than one cubic mile.

2. 4. All solid gold cubes are smaller than one cubic mile.

What I said about 1 and 2 applies here as well, I find. If we bump into a gold cube this large, we are surprised—and start thinking there might well be further ones. If we stumble upon a uranium cube of this size, we are surprised again. But we find our reasons for thinking that such a cube cannot exist unaffected and will instead start investigating this extraordinary case (if it persists for long enough). As far as I see, the difference between the shaky and the persistent attitude applies as well to the other examples prominent in the literature.Footnote ¹²

I am well aware that this sounds at best partially convincing. I am deliberately painting in black and white here in order to elaborate the opposition between the persistent and the shaky attitude. Obviously, one would be prepared to say how one would respond in such cases only if they would be described in much more detail, especially concerning the evidence that led one to believe in the relevant generalizations in the first place. So, there is also a lot of grey.

There are at least two different kinds of grey. First, there is a broad range of attitudes between the two extremes I have described. Being shaky means to be very shaky; the belief in further positive instances may instead fade more slowly. And, being persistent means to be strictly persistent; the belief in further positive instances may instead fade so lately that we never come to the point of testing it. Second, if confronted with such cases, we would in a sense widen our perspective. Take the uranium cube again. If we would really bump into such a large uranium cube, we would not simply mumble ‘Impossible!’ and stick to the belief that there will be no further exceptions. Rather, we would say that our original law was qualified by a ceteris paribus clause, anyway, and that a thorough investigation of the case will allow us to get clearer about normal and exceptional conditions. However, as fascinating as it is, the issue of ceteris paribus laws is certainly beyond the scope of this paper.Footnote ¹³

There are now two ways to respond. One may either say there is too much grey not decomposable into black and white. Or, one may say that there is an important insight in my black and white distinction which opens a fruitful way to analyze the shades of grey. I hope I have given at least some plausibility to proceeding on the second response.

If this is the right way to see the matter, treating a generalization strictly as a law is really to take the strictly persistent attitude towards it. This conclusion leads us to a further consequence, namely that the characteristic of lawlikeness is not something to be found in the propositional content of the generalization; rather, it lies in our inductive attitude towards it or its instantiations. This consequence will be of crucial importance in the sequel.

The account given so far is obviously very close to the old idea that laws are not general statements, but rather inference rules or inference licenses. The idea goes back at least to Ramsey ([Reference Ramsey and Mellor1929] 1978), who stated it very clearly: “Many sentences express cognitive attitudes without being propositions; and the difference between saying yes or no to them is not the difference between saying yes or no to a proposition” (135ff.). And, “… laws are not either, [namely propositions]” (150). Rather: “The general belief consists in (a) A general enunciation, (b) A habit of singular belief” (136). The idea has become quite popular among philosophers.

From a purely logical point of view, however, it is hard to see the difference between accepting the generalization as an axiom and accepting the corresponding inference rule for each instantiation. The only difference is that the rule is logically weaker; the rule is made admissible by the axiom, but the axiom cannot be inferred with the help of the rule. It has been little explained what else beside this unproductive logical point could be meant by the slogan ‘laws are inference rules’.

Still, one might say that the inference-license perspective emphasizes the single case. This emphasis has now been stripped of its merely rhetorical character; it is reflected, I think, in my central notion of persistence and thus finds a precise induction-theoretic basis. In this perspective, the mark of laws is not their universality which breaks down with a single counter-instance, but rather their operation in each single case, which is not impaired by exceptions. Here, my account meets with Cartwright (Reference Cartwright1989) and her continuous efforts to explain that physical laws are deceptive and that we should rather attend to the single case and to the capacities (co)operating in it. In Spohn 2002 the point is argued a bit more extensively.

So much for some striking agreements. The most obvious disagreement is with Popper, of course. There is no doubt about how much philosophy of science owes to Popper. In view of this, my account is really ironic, since its conclusion is, in a way, that the mark of laws is their not being falsifiable by negative instances; only accidental generalizations are subject to such falsification. To be a bit more precise: Of course, any generalization is falsified by a single counterinstance. But, falsified generalizations are to be rejected according to Popper. By contrast, I have argued that the belief in the further instances is shattered by the falsifying instances only in the case of accidental generalizations, but not in the case of laws. No doubt, the idea that the belief in laws is not given up so easily is familiar at least since Kuhn Reference Kuhn1962, and already Popper (Reference Popper1934, chap. IV, sec. 22) has insisted that the falsification of laws proceeds by more specialized counterlaws rather than by mere counterinstances. Here, however, the point is boiled down to its induction-theoretic essence.

5. Laws and Enumerative Induction

There is a striking and severe tension between Sections 3 and 4. We saw that, given symmetry, PIR is not feasible. Hence, we retreated to NNIR as a preliminary explication of enumerative induction. Then we noticed that enumerative induction applies only to laws. Finally, I have proposed an explication of laws according to which instances are independent of each other; this is what persistence amounts to. Thus, we arrived at complete instantial irrelevance that is rather a caricature of NNIR and not in agreement with enumerative induction at all. Something must have gone badly wrong.

No, there is only a subtle confusion. Belief in a law is more than belief in a proposition. It is a certain doxastic attitude, and that attitude as such is characterized by the independence in question: If I would have just this attitude—just the belief in a strict law and no further belief—my κ would exhibit this persistence or independence. Enumerative induction, by contrast, is not about what the belief in a law is, but about how we may acquire or confirm this belief. The two inductive attitudes involved may be easily confused, but the confusion cannot be identified as long as one thinks that belief in a law is just belief in a proposition.

However, what could it mean to confirm a law if it does not mean to confirm a proposition? My definition of confirmation in Section 2 applies only to the latter. Hence, the talk of the confirmation of laws, i.e., of a second-order inductive attitude towards a first-order inductive attitude, is so far merely metaphorical. Can we do better?

Yes, we can. There is fortunately clear precedent in the literature. Given the close similarity between probability and ranking theory, one might notice that a law as I conceived it, is nothing but a sequence of independent, identically distributed random variables translated into ranking terms. It thus becomes obvious that de Finetti (Reference de Finetti, Kyburg and Smokler1937) addressed exactly our problem in the probabilistic context. In his celebrated theorems, de Finetti showed that there is a one to one correspondence between symmetric probability measures for an infinite sequence of random variables and mixtures of Bernoulli measures according to which the variables are independent and identically distributed; and he showed that the mixture focuses more and more on a single Bernoulli measure as evidence accumulates. He thus showed to the objectivist that subjective symmetric measures provide everything he wants: beliefs about statistical hypotheses that converge toward the true one with increasing evidence.

De Finetti's issue between objectivism and subjectivism is not my concern. Ranking functions are thoroughly epistemological and have as such no objective interpretation.Footnote ¹⁴ Still, we can immediately translate de Finetti's theory into an account of the confirmation of laws as conceived here. I find the basic construction illuminating, despite its formalistic appearance.

Let us return to our simple one-property frame. We had two universal generalizations G ₁ and G ₀. But, there are infinitely many persistent, lawlike attitudes. If we define for all r, $s\in \mathbf{N}^{+}$ ,

and

then $\Lambda =\{ \lambda _{t}\vert t\in \mathbf{Z}^{+}\} $ includes all and only the persistent attitudes (where $\mathbf{Z}^{+}=\mathbf{Z}\cup \{ \infty,\,{\mbox{$-$}} \infty \} $ ). Λ contains precisely the ‘Bernoullian’ ranking functions which are symmetric and according to which each instance is independent from all others. For $t> 0$ λ_t believes in G ₁ and disbelieves in each negative instance with rank t. For $t< 0$ it is just the other way around; such a λ_t believes in G ₀ and disbelieves in each positive instance with rank t. In short, each λ_t counts the number of counterinstances within $\{ \boldsymbol{x}\,\mathrm{takes}\,\,z_{1},\,\ldots,\,z_{n}\} $ to the generalization it believes in and multiplies it by t (or −t).

What, then, is the difference between, e.g., λ₁ and λ₂? There is none in content and none in persistence. The only difference lies in the disbelief in negative instances; λ₂ is a firmer law, one might say, than λ₁. Rather, for technical reasons, we have to include λ_∞ and λ_−∞. λ₀, finally, does not represent a law at all. Rather, it represents lawlessness, indeed complete agnosticism; nothing (except the tautology) is believed in λ₀. Its special role will be discussed in the final section.

Now, believing in laws, confirming and falsifying laws, etc., are doxastic attitudes towards laws, which will here be modeled, of course, by a ranking function ρ over the set Λ of possible laws. If the possible laws are possible first-order attitudes, then ρ is a second-order attitude, which, however, induces a first-order attitude. What, according to ρ, is the rank of a proposition $A\subseteq W$ , i.e., the degree of disbelief in A? It is the minimum of all the disbeliefs in A, according to the possible laws in Λ, weighed by the disbelief in the laws according to ρ; that is, the first-order attitude induced by ρ is just the mixture of Λ by ρ as defined in Definition 5.

Are we talking about a specific second-order attitude ρ? No, you may have any ρ you like. The following considerations are perfectly general in this respect. Let us call ρ proper, though, iff at most one of ρ(λ_∞) and ρ(λ_−∞) is finite. Now we can start translating de Finetti's theorems.

First, we have

Second, we have: For each regular symmetric ranking function κ satisfying NNIR, there is a proper ranking function ρ over Λ such that κ is the mixture of Λ by ρ. We may indeed strengthen the claim. Suppose we mix, e.g., λ₁ and λ₂ by some ρ with $\rho (\lambda _{1}) =\rho (\lambda _{2}) =0$ . Then λ₂ is obviously a redundant component of the mixture; it never determines the result of the mixture, i.e., the relevant minimum. Because of such redundant components mixtures are never unique.Footnote ¹⁵ Uniqueness can be achieved only with minimal mixtures, as we might call them. However, we must be careful in catching the right kind of minimality. The point of the following definition will become fully clear only with Theorem 5 below.

Definition 8. $\lambda _{s}$ is a redundant component of the mixture of Λ by ρ wrt a proposition A iff there is no proposition B such that

i.e., iff $\lambda _{s}$ does not determine the value of the mixture for any $A\cap B$ . $\lambda _{s}$ is a strongly redundant component of the mixture of Λ by ρ iff λ_s is a redundant component of the mixture wrt to all $A_{m,n}$ $(m,\,n\geq 0) $ , where $A_{m,n}$ is the proposition that (in some order) m of the first $m+n$ objects have P and the other n objects lack P. Finally, the mixture of Λ by ρ is called minimal iff for all its strongly redundant components $\lambda _{s}$ $\rho (\lambda _{s}) =\infty $ .

Hence, in a minimal mixture, all strongly redundant components get weight ∞ and cannot enter the mixture at all. The strengthened claim is then:

Proof. Let κ be a regular symmetric function satisfying NNIR, let f be its representative function forming an infinite triangle of nonnegative integers, and let $c=\mathrm{sup}\,f$ be the supremum of f, which may be finite or infinite. Let us focus on simple paths starting at the boundary of the triangle and making no turns. These paths take two forms. For each $m\geq 0$ there is the right path $f(m,\,0) $ , $f(m,\,1) $ , $f(m,\,2) $ —starting at the left and going always right—and for each $n\geq 0$ there is the left path $f(0,\,n) $ , $f(1,\,n) $ , $f(2,\,n) $ —starting at the right and going always left. We know that the simple paths are nondecreasing (like all others). NNIR entails, moreover, that the simple paths do not accelerate; whenever i, j, k are three consecutive members of such a path, then $k-j\leq j-i$ .

Each simple path either goes to infinity or reaches a maximum and then remains constant. Let us define $a_{m}$ to be the supremum of the mth right path $f(m,\,0) $ , $f(m,\,1) $ , …; and $b_{n}$ to be the maximum of the nth left path $f(0,\,n) $ , $f(1,\,n),\,\ldots $ $(m,\,n\geq 0) $ . Again, both sequences $\boldsymbol{a}=(a_{0},\,a_{1},\,\ldots) $ and $\boldsymbol{b}=(b_{0},\,b_{1},\,\ldots) $ must be nondecreasing, and due to NNIR, also nonaccelerating. Either $a_{0}=0$ or $b_{0}=0$ or both, and $c=\mathrm{sup}\,\,\boldsymbol{a}=\mathrm{sup}\,\,\boldsymbol{b}$ .

With the help of the two sequences $\boldsymbol{a}$ and $\boldsymbol{b}$ , we can now construct the relevant minimal mixture ρ. If $a_{1}-a_{0}\mathrel{\mathop:}= r$ , we set $\rho (\lambda _{-r}) =a_{0}$ ; and if $a_{m}$ is any point at which $\boldsymbol{a}$ decelerates, i.e., such that $a_{m}-a_{m-1}> a_{m+1}-a_{m}\mathrel{\mathop:}= r$ , we set $\rho (\lambda _{-r}) =a_{m}-mr$ . Similarly, if $b_{1}-b_{0}\mathrel{\mathop:}= s$ , we set $\rho (\lambda _{s}) =b_{0}$ ; and if $b_{n}$ is any point at which $\boldsymbol{b}$ decelerates, i.e., such that $b_{n}-b_{n-1}> b_{n+1}-b_{n}\mathrel{\mathop:}= s$ , we set $\rho (\lambda _{s}) =b_{n}-ns$ . If for any $t\in \mathbf{Z}^{+}$ $\rho (\lambda _{t}) $ is not thereby defined, we set $\rho (\lambda _{t}) =\infty $ . This completes the construction of ρ. Note, in particular, that this entails $\rho (\lambda _{0}) =c$ . Hence, the lawless λ₀ is a relevant component of the mixture only if c is finite.

Since either $a_{0}=0$ or $b_{0}=0$ , there is some $t\in \mathbf{Z}^{+}$ with $\rho (\lambda _{t}) =0$ . Since at least one of a ₁ and b ₁ is finite, either $\rho (\lambda _{\infty }) =\infty $ or $\rho (\lambda _{-\infty }) =\infty $ or both. Hence, ρ is a proper ranking function over Λ.

The mixture of Λ by ρ indeed generates the representative function f: For all m, $n\geq 0$ we have either $f(m,\,n) =a_{m}$ or $f(m,\,n) =b_{n}$ , since either $f(m,\,n+1) =f(m,\,n) $ or $f(m+1,\,n) =f(m,\,n) $ , and thus either the right or the left simple path through $f(m,\,n) $ does not increase after $f(m,\,n) $ . Now suppose $f(m,\,n) =b_{n}\leq a_{m}$ , and let us check whether our mixture yields the same result:

As above, let $A_{m,n}$ be the proposition that (in some order) m of the first $m+n$ objects have P and the other n objects lack P. Hence,

How to calculate this minimum? Let $b_{n^{*}}$ be the largest point before $b_{n}$ , where $\boldsymbol{b}$ decelerates, and let $s^{*}=b_{n^{*}+1}-b_{n^{*}}$ . Hence, $b_{n^{*}}=b_{n}-s^{*}(n-n^{*}) $ . What about $ns^{*}+\rho (\lambda _{s^{*}}) $ ? We have:

Moreover, it is clear from the construction that $ns+\rho (\lambda _{s}) \geq ns^{*}+\rho (\lambda _{s^{*}}) $ for all other $s\in \mathbf{N}^{+}$ . The same reasoning shows that $mr+\rho (\lambda _{-r}) \geq a_{m}\geq b_{n}$ for all $r\in \mathbf{N}^{+}$ . Thus, indeed, $f(m,\,n) =b_{n}$ according to the mixture. Of course, if $f(m,\,n) =a_{m}\leq b_{n}$ , the corresponding argument holds.

Some $\lambda _{t}$ ( $t\in \mathbf{Z}^{+}$ ) receiving finite rank by ρ may be a redundant component of the mixture of Λ by ρ wrt A _0,0 (the tautology); this always happens when two successive members $a_{m}$ and $a_{m+1}$ of $\boldsymbol{a}$ or $b_{n}$ and $b_{n+1}$ of $\boldsymbol{b}$ are points of deceleration. But, none of them is strongly redundant, and the mixture is indeed minimal in the sense of Definition 8. This, however, will become clear only with the next theorem. It will also be obvious, then, that the ρ we have constructed is unique, i.e., it provides the only minimal mixture generating the representative function f. □

The final step in our translation of de Finetti is to inquire how the mixture is changed by evidence. This can be directly read off from the results above. Suppose that we collect the evidence $A_{m,n}$ that m of the first $m+n$ objects have P and the other n objects lack P. If we start with the regular symmetric κ with representative function f, what is then the a posteriori ranking function $\kappa _{m,n}$ on the space of possibilities for the infinitely many remaining objects? Well, we learn by conditionalization; hence, for any proposition B within this space $\kappa _{m,n}(B) =\kappa (B\vert A_{m,n}) $ . The representative function $f_{m,n}$ of $\kappa _{m,n}$ is thus given by $f_{m,n}(p,\,q) =f(m+p,\,n+q) -f(m,\,n) $ .

Now, suppose that κ is the minimal mixture of Λ by ρ. What, then, is the unique $\rho _{m,n}$ so that $\kappa _{m,n}$ is the minimal mixture of Λ by $\rho _{m,n}$ ? We know that f is the result of the mixture by ρ, i.e.,

Thus, we have for all p, $q\in \mathbf{N}$ :

This already suggests how to define $\rho _{m,n}$ . However, $\rho _{m,n}$ has to be a minimal mixture; therefore, we still need to eliminate some of the components originally having finite rank. For this purpose, let $a_{m^{*}}$ be the largest member of a up to a _m, where a decelerates and $b_{n^{*}}$ is the largest member of b up b _n, where b decelerates (thus, possibly $a_{m^{*}}=a_{m}$ and $b_{n^{*}}=b_{n}$ ), and let $r^{*}=a_{m^{*}+1}-a_{m^{*}}$ and $s^{*}=b_{n^{*}+1}-b_{n^{*}}$ . Now we can state

Theorem 5. Define for r, $s\in \mathbf{N}^{+}$ :

and

and if $r> r^{*}$ and $s> s^{*}$ , then $\rho _{m,n}(\lambda _{-r}) =\rho _{m,n}(\lambda _{s}) =\infty $ , then $\kappa _{m,n}$ is the minimal mixture of Λ by $\rho _{m,n}$ .

The point of defining minimality as we did in Definition 8 now becomes clear. As mentioned, some components of the mixture of Λ by ρ may be initially redundant, i.e., wrt to A _0,0. Still, they may become nonredundant after conditionalization by $A_{m,n}$ . Hence, they have to be included already in the original mixture. Otherwise, we could not have obtained $\rho _{m,n}$ from ρ so easily as in Theorem 5.

The theorem has three important consequences. First, it helps to reestablish positive instantial relevance. Suppose, we find the $m+n+1$ object to have P; thus, our evidence increases from $A_{m,n}$ to $A_{m+1,n}$ . How does the mixture change from $\rho _{m,n}$ to $\rho _{m+1,n}$ ? Insofar $\rho _{m+1,n}$ is finite we have for r, $s\geq 1$ :

and

Hence, in any case,

That is, each λ_−r as opposed to any of the λ_s is more disbelieved in $\rho _{m+1,n}$ than in $\rho _{m,n}$ (by r ranks). In other words, the additional positive instance is positively relevant to the positive lawlike attitudes. So, on the level of the second-order attitudes we indeed have exceptionless positive instantial relevance, which is blurred, though, by the mixture and thus weakens to NNIR on the level of first-order attitudes. Theorem 2 has shown that this weakening is unavoidable, but now we see that it is only an artifact of the mixture.

This observation teaches us, secondly, that as more and more positive instances accumulate and $m-n$ diverges to infinity, $\rho _{m,n}(\lambda _{-r}) -\rho _{m,n}(\lambda _{s}) $ (r, $s\geq 1$ ) diverges to infinity as well, i.e., the disbelief in the negative lawlike attitudes heads for infinite firmness. This parallels de Finetti's observation in the probabilistic case.

So, all in all, we have seen that de Finetti's account of the confirmation of statistical hypotheses may be perfectly translated into ranking theoretic terms, thus deepening our understanding of enumerative induction and lawlikeness.

There is still a third lesson, which has in fact no probabilistic analogue. Thus, it goes a little step beyond de Finetti and deserves a brief concluding section of its own.

6. The Apriority of Lawfulness

This lesson concerns the special role of λ₀. We noticed already that λ₀ is total agnosticism expressing lawlessness instead of lawfulness. Now, we either have $\rho (\lambda _{0}) =\infty $ , which entails $\rho _{m+n}(\lambda _{0}) =\infty $ for all m, $n\in \mathbf{N}$ . Then, ρ embodies the maximally firm belief that some law or other will hold. This belief would indeed be invariable, not refutable even by very long sequences of apparent random behavior of the instances with respect to P. This does not appear reasonable.

The alternative is that we give ρ(λ₀) some finite value; hence, $\rho _{m,n}(\lambda _{0}) =\rho (\lambda _{0}) -f(m,\,n) $ . This entails that with each unexpected realization of an instance λ₀ gets less disbelieved. After too many disappointments, we shall eventually have lost our belief in lawfulness and any belief about the behavior of new objects concerning P, the belief in lawlessness being the only remaining option. This may also sound implausible. However, ρ(λ₀) may be very large so that the agnostic state is in fact never reached.

The more relevant observation, though, is that the whole story I have told about the single property P can be generalized to any finite number of properties P ₁, …, P _m in a straightforward way. We can define Carnap's Q-predicates, i.e., the atoms of the Boolean algebra of properties generated by P ₁, …, P _m; for each Q-predicate Q _k, we can consider the generalization ‘There is no Q _k’ and the corresponding laws, i.e., persistent attitudes; all the theorems of Section 5 continue to hold. So, what we would really do, if lawlessness with respect to P threatens, is to try to correlate P with some other properties and to pursue the investigation within a larger space of properties.

Within such a larger space, more complex forms of laws become available, going beyond persistent attitudes towards ‘There is no Q _k’. As already mentioned, the ranking theoretic framework in particular allows of an analysis of ceteris paribus laws (cf. Spohn Reference Spohn2002, sec. 4). So, there are rich prospects of generalization. I don't know, though, whether and how the de Finettian story I have told concerning simple laws (about P or the Q _k) carries over to such more complex laws. And I don't know of any working account of conceptual change answering the threat of lawlessness within any given set of properties or conceptual framework. So, there is still a lot to do as well.

However, let me finally emphasize what my brief discussion of λ₀ means in more traditional terms. Kant tried to overcome Hume's objectivity skepticism generally with his transcendental logic and its synthetic principles a priori, and Hume's inductive skepticism particularly with his a priori principle of causality. This principle ascertained rather only the rule- or law-guidedness of everything happening and was thus also called the principle of uniformity of nature (cf., e.g., Salmon Reference Salmon1966, 40ff.). As was often observed, this principle did not offer any constructive solution to the problem of induction, since it does not give any direction as to specific causal laws or specific inductive inferences. Still, it provided, if a priori true, an abstract guarantee that our inductive efforts are not futile in principle. Is it a priori true?

Nowadays, two notions of apriority are usually distinguished. A proposition is unrevisably a priori if it must be believed and cannot be given up under any evidential circumstances. This is certainly the notion that Kant used, though he did not express it in this way, and which Quine attacked when attacking analyticity. By contrast, a proposition is defeasibly a priori if it is to be believed initially, prior to any experience (and may be given up later on). The prior probabilities discussed by Bayesians are a paradigm of defeasible apriority because they, of course, are expected to change.

Now, our initial ranking function is some regular symmetric κ satisfying NNIR. Via Theorem 4, κ uniquely corresponds to some ranking function ρ over Λ. The belief in lawfulness, then, is the same as the disbelief in lawlessness, i.e., $\rho (\lambda _{0}) > 0$ . We saw that this as an extremely reasonable assumption. And, we now see that it is tantamount to the defeasible apriority of lawfulness: We must start by believing in the uniformity of nature.

The unrevisable apriority of lawfulness, however, is expressed by the stronger condition $\rho (\lambda _{0}) =\infty $ . We also saw that this condition does not appear reasonable, at least if one relates it to the property P or, more generally, to any fixed set of properties. Still, it may be unrevisably a priori that there is some set of properties with respect to which nature is uniform. I am not prepared to decide whether or not the unrevisable apriority of lawfulness is defensible in this sense. But, I think the issue is more clearly arguable on the basis provided here.