Popper’s “generating conditions” (which may be regarded as the partial causes of the actual outcome, see §6) may be regarded, then (according to §4), as primarily the physical origins of the physical probabilities, being related only derivatively with our descriptions of the experimental set-up. We may conjecture that only certain, repeatable aspects of reality are probabilistically relevant (e.g. the f); and of course, such aspects would have to be described in the course of making such a conjecture, and be related to our descriptions of the experimental set-ups during the testing of such conjectures—but clearly that relation need not (if we are Realists about such things) be identity.
Furthermore, when tossing a die the propensities (which are properties of the generating conditions) are, for Popper (1983: 359), “not properties inherent in the die [… but] relational properties of the total objective situation.” E.g., the physical probabilities for the throw of a die, say D, that had been biased by having some metal put under one of its faces, might depend upon how it was tossed, upon what it was tossed onto (e.g. sand or marble), upon how the Earth’s magnetic field was varying as it was tossed, and so forth (and note that, if we are Realists about such things, some nomically relevant aspects of the generating conditions may well be indescribable).
It may therefore be a confusion of two kinds of descriptions—our conjectures about the underlying physical states, and the specifications of our experiments—that explains why Salmon (1979) thought that Popper faced the following reference class problem.
This is the problem from which Popper’s discussion takes its departure—how can we assign a probability other than ¼ to the result 6 when the unbiased die is thrown? Popper’s chief criticism of the frequentist position is that it seems to lead to two distinct values, depending upon which sequence you choose as the reference class. Now, Popper tells us, the single case probability statement is “about certain properties of the experimental conditions”. But which among all the properties of an experimental set-up are the “certain properties” to which the probability is relative? Are we talking about the toss of any old die, biased or unbiased? or about any method of tossing? or tossing by means of a dice cup? or by a left-handed person? or from a certain height above the table? or with a closely specified angular momentum? (Salmon 1979: 197)
Since Gillies (2000: 119-23) recently reiterated this problem, as a reason for taking a long-run approach, I shall consider it in a bit more detail. It is claimed that, since the probability of a single event varies with the description of the generating conditions that the event is instantiating, hence such probabilities should be associated primarily with the classes defined by those descriptions, and only derivatively with the individual events. But clearly, Popper’s generating or “experimental conditions” are the physical tosses in all their actual detail, however we might describe them.
Popper asked us to imagine that we had a well-defined sequence of tosses, b, amongst which the tosses with the unbiased die were undetectable (e.g. perhaps they were all tossed by means of a particular dice cup), and so because the physical conditions that generate the physical probabilities might be peculiar to each toss (see below), hence such a description (e.g. tossed by means of a particular dice cup) would not describe a “generating condition” (cf. §4). Such descriptions are of course useful experimentally, but why should we not also hypothesise about the underlying sources of indeterminism, and thence (if we are Realists about such things) consider the net single-case propensities that they would give rise to? It does seem that Popper was doing just that.
As far as I can tell, there might be several sources of indeterminism in a typical toss—e.g., chemical reactions involved in our tossing the die (about which we are ignorant as we toss it) and in its interactions with its environment, chaotic amplification of background indeterminism (e.g. in the air through which it rolls), our personal input (rather more mysteriously), and so forth—whence the net chanciness could vary considerably even between tosses of the same die by the same (ordinarily) describable mechanism.
Anyway, since Salmon (1979: 195) believed that “Reichenbach had addressed precisely the problem Popper poses,” the rest of this section will consider some substantial differences. Reichenbach had indeed said the following:
[When] we produce a
probability sequence by throwing a die, we demand that each throw be played
with the same probability, that is, there should not be occasional exceptions
where a loaded die is used
(Reichenbach 1971: 168)
But of course, if Reichenbach had only meant that for a sequence of throws that was defined to be the throws of some die, we would need to use that die and not another, then although that would be true it would be trivial. Similarly, were the phrase ‘the same probability’ to mean that the same description was to be satisfied, it would be trivial to ensure that Reichenbach’s demand was never satisfied (e.g. were the description to include the times of the throws) or that it was always satisfied (e.g. were it a complete description of that sequence).
Essentially, that is the reference class problem for Frequentism. E.g. consider a large number of very similar throws of the biased die D (above)—if the Earth’s magnetic field varied for some of the throws, but we had not included something like ‘constant magnetic field’ in our description, then although all the throws will have satisfied our description, some will really have been made with a different bias.
Reichenbach is a Frequentist, who identifies probabilities with the limits of frequencies within infinite sequences (that idealise relatively long empirical sequences), so in order to demand that the throws are played with the same probability he considers (1971: 169) each throw to be in yet another infinite sequence—a so-called virtual sequence, which we may envisage as progressing vertically, if the original sequence of actual throws is envisaged horizontally—and then demands the equality of the limits of those virtual sequences. Now, precisely what that that amounts to depends upon what the virtual sequences are composed of.
Were exactly the same throw repeated, each virtual sequence would, in a deterministic world (such as Salmon thinks our analysis ought to be able to accommodate), contain only the one result. But then all the probabilities in the virtual sequence would be either 1 or 0, depending upon the actual outcome of that throw, rather than upon whether or not they were made with a differently loaded die. And in an indeterministic world (such as this one appears to be), the singular probabilities could vary a lot between apparently similar throws of the same die (as above), so that very few of the throws would be “played with the same probability.” And then the decisive factor would again not be the loading of the die.
So presumably it is only the kind of throw that is repeated, whence (since Reichenbach would define such kinds by their descriptions) the reference class problem for Frequentism would again arise. But anyway, since the equality of the probabilities throughout the original sequence just means (for Reichenbach) that each virtual sequence should possess the same limit, hence we might specify that each virtual sequence be the same, from that throw onwards, as the original sequence, in which case the singular probabilities of the original sequence would be all too trivially equal.
In fact, Reichenbach (1971: 374) thought that we might take the reference class for a singular probability to be, in general, the narrowest class for which reliable statistics exist. But not only might several such classes be relevant, without there being reliable statistics for their intersection, such an approach makes singular probabilities depend upon the state of our knowledge. So it either outlaws, or just fails to address, the objective probabilities that concerned Popper.
Reichenbach (1971: 375-6) did suggest that a singular probability might be imagined (more objectively) to be the limit of limits for classes that become narrower and narrower, but he gave no reason for supposing that such classes could always be given in enough ways for such limits to exist, nor any reason for supposing that those limits would be unique. And anyway, for Reichenbach (1971: 377) all singular probabilities are fictional, as he emphasised by giving the following argument against them.
To state the argument briefly: if probability has something to do with the reliability of predictions, the probability statement must be verifiable in terms of the occurrence of the event predicted; otherwise the statement will be empty so far as predictions are concerned. The frequency interpretation satisfies this condition inasmuch as it verifies a degree of probability through repeated occurrence of the event. If, however, the meaning of the probability statement refers to a single event, it is impossible to verify the statement in terms of the occurrence of the event; and therefore the statement has no predictional value. (Reichenbach 1971: 371)
But were we to work out the half-life of a uniquely instantiated isotope via the half-lives of more common isotopes, correlating them with some other attributes that are directly observable, then surely that would have “predictional value.”
Furthermore, because all of Reichenbach’s probabilities are idealisations (as were von Mises’), it is quite obscure what his claim that all singular probabilities are fictional amounts to. Presumably it amounts to the fundamental difference between him and Popper, but unfortunately a further indication of the obscurity here is that he (1971: 348) also said that, since his theory was approximately correct in practice, he had given the meaning of physical probability—even though he had just acknowledged the following argument:
A limit at a given value p
is compatible with every finite beginning of the probability sequence; since we
can count the frequency only in a finite initial section, all limit statements
must be called nonverifiable and, consequently, meaningless. (Reichenbach 1971: 347)
Of course, if the finite frequencies do have the real meaning then he ought to be classed with the more easily dismissed Finite Frequentists (of §2). After all, consider again the few throws of the fair die in Popper’s thought-experiment—surely Reichenbach ought to regard the real probability (of getting a 6) for those few throws as the actual frequency (of 6) within b.