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 ρ(λ0) = ∞, which entails ρm+n(λ0) = ∞ for all m,n ∈ N. Then ρ embodies the maxi-mally firm belief that some law or other will obtain. 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, ρm,n(λ0) = ρ(λ0) – 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 be-lief in lawfulness and any bebe-lief 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 P1, ..., Pmin a straightforward way. We can define Carnap’s Q-predicates, i.e., the atoms of the Boolean algebra of properties generated by P1, ..., Pm; for each Q-predicate Qk we can consider the generalization “there is no Qk” and the corresponding laws, i.e., persistent attitudes; and then all the theorems of section 4 continue to hold. So, what we would really do if lawlessness with respect to P threatens is to try to cor-relate P with some other properties and to pursue the investigation within a larger space of properties.
Within such a larger space also more complex forms of laws become available going beyond persistent attitudes towards “there is no Qk”. As already mentioned, the ranking theoretic framework in particular allows of an analysis of ceteris pari-bus laws (cf. Spohn 2002, sect. 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 Qk) 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 gen-erally with his transcendental logic and its synthetic principles a priori and Hume’s inductive skepticism particularly with his a priori principle of causality. This prin-ciple ascertained rather only the rule- or law-guidedness of everything happening and was thus as well called the principle of uniformity of nature (cf., e.g., Salmon 1966, pp.40ff.). As was often observed, this principle did not offer any constructive solution of the problem of induction, since it does not give any direction as to spe-cific causal laws or spespe-cific 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 evi-dential circumstances. This is certainly the notion which Kant used, though 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 are, of course, ex-pected 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. ρ(λ0) > 0.
We saw that this is an extremely reasonable assumption. And we now see that it is tantamount to the defeasible apriority of lawfulness: we must start believing in the uniformity of nature.
The unrevisable apriority of lawfulness, however, is expressed by the stronger condition ρ(λ0) = ∞. 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 prop-erties. 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.
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