I propose to interpret the objective probability of a single event as a measure of an objective propensity—of the strength of the tendency, inherent in the specified physical situation, to realize the event—to make it happen. (Popper 1983: 395)
Popper’s famous argument for his propensity interpretation of physical probability (from the fifties, repeated in 1983: 352-6) concerned an indefinitely extended sequence, b, of dice-throws, most of which had been made with a loaded die for which prob(6) = ¼ had been determined empirically, but one or two of which had been made (at unknown locations within b) with a symmetrical and homogeneous (and therefore presumably fair) die.
Those few throws with the latter die are members of a (hypothetical, or fictionally actual) sequence of throws, b, of which ¼ are 6s, yet the objective probability of getting a 6 in those few cases is clearly not ¼, but 1/6. Even a Frequentist would conjecture that 6 would turn up 1/6 of the time in a (fictionally virtual, cf. §5) sequence of throws, c, made with the fair die (even though that die is actually, we may imagine, only ever thrown within b). So although those few fair throws are members of both b and c, which are both indefinitely long sequences, we are all quite sure that, for those throws, the true probability of 6 is 1/6, not ¼.
Popper then (1983: 355-6) explained why that is such a problem for the Frequentist, as follows.
[The Frequentist] is forced to introduce a
modification of his theory—apparently a very slight one. He will now say that
an admissible sequence of events (a reference sequence, a ‘collective’) must
always be a sequence of repeated conditions. Or more generally, he will say
that admissible sequences must be either virtual or actual sequences which are
characterized by a set of generating conditions—by a set of conditions whose
repeated realization produces the elements of an independent sequence.
If
this modification is introduced, then our problem is at once solved [… Also] it
seems that what I have here described as a ‘modification’ only states
explicitly an assumption which most frequency theorists (myself included) have
always taken for granted. Yet if we look more closely at this apparently slight
modification, we find that it amounts to a transition from the frequency
interpretation to the propensity interpretation.
The
frequency interpretation always takes probability as relative to a sequence
which is assumed as ‘given’; and it works on the assumption that a probability
is a property of some given sequence. But with our modification, the
sequence in its turn is defined by its set of generating conditions; and
in such a way that probability may now be said to be a property of the
generating conditions. […]
[Hence]
the singular event a possesses a probability p(a, b)
owing to the fact that it is an event produced, or selected, in accordance with
the generating conditions […] In this way, a singular event may have a
probability even though it may occur only once; for its probability is a
property of its generating conditions: it is generated by them.
[Hence] the virtual frequency must also depend upon these conditions. But this means that we have to visualize the conditions as endowed with a tendency, or disposition, or propensity, to produce sequences with frequencies equal to the probabilities; which is precisely what the propensity interpretation asserts. (Popper 1983: 355-6)
That tendency (to produce sequences with certain frequencies) is presumably a consequence of the repetition of the generating conditions (each instantiation of which would yield a probability). That is, Popper’s neo-classical propensities seem to be analogous to the proportions inside Laplace’s urn—cf. how, as a consequence of repeating random draws from Laplace’s urn (in §1), the limit frequency of white balls drawn from it was almost certain to equal the proportion of white balls within it.
Note that, were the balls in the urn (i.e. the possible outcomes) replaced by points in a continuum, the classical definition of probability (see §1) would become a normalised measure (of those favourable possibilities) like Kolmogorov’s (see §1). While Frequentism seems unable to extend the classical definition coherently (§3, also §5), the standard axioms of probability (or some post-paradoxical refinement of them, see the last 3 sections) allow even the complex numbers of quantum mechanics to be accommodated, with the favourable futures being modelled as points in a unit continuum of all the physically possible futures. That arrangement of possible futures as a continuum (rather than as balls placed randomly inside an urn) is not ad hoc because the observable magnitudes (which usually form the horizontal axis of the graph of a probability distribution, e.g. the possible positions of detected particles in the two-slit experiment described below) presumably vary continuously, and there is no reason why the arrangement of the physically identical possible worlds upon that continuum of the various possibilities (i.e. below the graph of the probability distribution) should differ in kind from that arrangement—especially since the two-slit experiment (see below) indicates that such probabilities (those vertical magnitudes) interact as objectively as force-fields do.
Anyway, were Popper’s shift to generating conditions (from empirical collections) indeed like shifting to the internal proportion of Laplace’s urn (from the results of draws from it) then it would be relatively clear how substantial Popper’s shift was, and how central the single case would then become—cf. how, even were more than one draw from the urn impossible for some reason, the internal proportion of balls would still determine the probability for that single draw.
Such an analogy would
therefore help to answer one of Salmon’s objections (1979: 196; see §5), that
Popper’s shift was merely terminological because it was a shift towards long-run (rather than single-case)
propensities, where that distinction is as follows (Gillies’ long-run approach
will be criticised in §8).
A long-run propensity
theory is one in which propensities are associated with repeatable conditions,
and are regarded as propensities to produce, in a long series of repetitions of
these conditions, frequencies which are approximately equal to the
probabilities. A single-case propensity theory is one in which
propensities are regarded as propensities to produce a particular result on a
specific occasion.
(Gillies 2000: 126; my emphasis)
Since most Frequentists had already assumed
Popper’s “modification” (as Popper admitted,
above)—e.g. von Mises’ collectives were collections of “uniform events” (1957: 12), or “repetitive
events” (1957: 15), while Reichenbach even accommodated talk of singular
cases (incoherently, see §5)—and since Popper was also a little unclear (see
below), hence it would indeed, were we to miss the force of such a realistic
analogy, be hard to tell whether Popper’s (1983: 356) idea (that the generating
conditions have a tendency to produce sequences) implies a long-run view of propensities
or not. But although Popper said that (1983: 397) “Propensities
are dispositions to produce frequencies,” and (1983: 358) “probabilities are conjectured or estimated statistical
frequencies in long (actual or virtual) sequences,” mostly he did seem
to mean single-case propensities, e.g.:
Thus the main difference between the frequency interpretation and the propensity interpretation lies in the status of singular probability statements. They play a peripheral role in the frequency theory but a central role in the propensity interpretation which sees, as it were, every single case as the outcome of a propensity, or perhaps of contesting propensities, even though these can be tested only statistically. (Popper 1983: 287; my emphasis)
Since a consideration of what Popper might have meant by “contesting propensities” will help to introduce §5, I will next consider some possibilities (before once again quoting Popper pro-single-case)—most trivially, when tossing a coin there is a tendency for heads and a complimentary tendency for tails, so such incompatible possibilities may have been intended. But perhaps he meant something like a coin that is biased towards heads being tossed by a conjuror biasing his tosses towards tails—while there would be one net propensity, for such a toss, the description is of two competing ones.
For Salmon (1979: 206), Popper’s propensities were primarily associated with descriptions (see §5), and Kyberg (1974: 361) had also noted an “unresolved tension” between Popper’s objectivity and how “we find that propensities are properties of conditions we intend to keep constant, of sequences which may be described as repetitions of an experiment.” But I think that it is more realistic to associate propensities with the physical states of the system, rather than with our descriptions of our experimental set-ups. Cf. fragility (since that concept will crop up again in §6), which, although it only exists as a concept because sequences of similar breakages have been observed, is presumably a consequence of the internal physical chemistry of each fragile item (for all that similar items would have similar internal structures).
Some evidence against that interpretation of Popper (not against that view of propensities) is that Popper (1983: 398) later wrote: “The propensity interpretation is, I believe, that of classical statistical mechanics.” However, he was there thinking of certain generating conditions being specified by descriptions (e.g., rolling a coin over a rough surface) that left unspecified the initial conditions (and hence the exact path of the coin), and those are deterministic scenarios (unless indeterminism is introduced via the relatively blind choice of how to role the coin), whereas Popper had earlier emphasised the importance of indeterminism (e.g. the following quote).
Anyway, when we are describing a quantum-mechanical set-up we are effectively conjecturing about the underlying reality, so it seems realistic to regard Popper’s “contesting propensities” as being such repeatable parts as we would conjecture would sum to our macroscopic scenarios (e.g. the particles’ f, it being the fact that particles come in natural kinds that presumably gives rise to the repeatability of our experiments).
The validity of the analogy with Laplace’s urn—i.e. the central role for single-case propensities—was emphasised again when Popper said that he was putting forward a new physical (or metaphysical) hypothesis:
It is the hypothesis that every experimental arrangement (and therefore every state of a system) generates propensities which can sometimes be tested by frequencies [… and] it is corroborated by certain quantum experiments. The two-slit experiment [decides] the issue against the purely statistical interpretation. (Popper 1983: 360)
The two-slit
experiment involves particles passing one at a time through a screen containing
two slits, towards an array of detectors. Since the observed frequencies do not
happen to be the sum of those for experiments in which only one slit is open,
hence something corresponding to the wavefunction f appears to be sufficiently
physical for it to interact with itself (after passing through both slits).
That is, the probability distribution |f|2 appears to be relevant in each
single case.
So, Popper seems to be
saying that physical states invariably give rise to (perhaps trivial)
propensities for future states, while the conjectured values of those
propensities can only sometimes be tested via empirical frequencies. Now, since
one of Gillies’ (2000: 119-24) three objections to single-case propensities was
illustrated via that two-slit experiment, I shall end this section by rejecting
that objection (for the other two, see §5 and §8).
Suppose that two scientists Mr A and Ms B are betting on where an electron will impinge in a particular repetition of the experiment. Mr A sets his probabilities equal to those calculated by the standard theory. Ms B, however, has noticed that there was a thunderstorm nearby, and knows from experience that the resulting electrical disturbances in the atmosphere often affect an experiment of this sort. She therefore adjusts her probabilities in the light of this factor. [So, it] seems better to analyse the singular probabilities in a particular repetition of the experiment as subjective probabilities rather than as objective probabilities exactly equal to the objective probabilities in a sequence of repetitions of the experiment. (Gillies 2000: 124)
By having the scientists bet on the outcome, a subjective interpretation is suggested, but we are here considering the metaphysical status of physical probabilities, so note that Gillies has not shown that such betting could not affect the objective probabilities (cf. §6). Furthermore, although Ms B’s probabilities may well be better than Mr A’s if her adjustments are correct, surely for them to be correct is for them to make her probabilities more like the true values for this particular set-up (thunderstorm and all)—and what could they be, if not the objective single-case ones? Section 5.