Abstract
I analyse critically what I regard as the most accomplished empiricist account of propensities, namely the long run propensity theory developed by Donald Gillies (2000). Empiricist accounts are distinguished by their commitment to the ‘identity thesis’: the identification of propensities and objective probabilities. These theories are intended, in the tradition of Karl Popper’s influential proposal, to provide an interpretation of probability (under a suitable version of Kolmogorov’s axioms) that renders probability statements directly testable by experiment. I argue that the commitment to the identity thesis leaves empiricist theories, including Gillies’ version, vulnerable to a variant of what is known as Humphreys’ paradox. I suggest that the tension may be resolved only by abandoning the identity thesis, and by adopting instead an understanding of propensities as explanatory properties of chancy objects.
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Notes
Gillies’ (2000, Ch. 6 and 7) suggests these advantages, but does not express them in the terms I use above, which is why I choose not to ascribe them to him verbatim in the text.
A complication is that the outcome event itself may only be identified relative to the experimental set up–indeed Popper thought that propensities were relational properties of the entire experimental set-up–but this does not alter the fundamental point.
Gillies is not always entirely clear on this point. Sometimes he seems to ascribe propensities to the set of repeated conditions relative to the sequence, rather than the other way around, as, for instance, when he notes: “The propensity theory claims that some sets of repeatable conditions have a propensity to produce in a long sequence of repetitions frequencies which are approximately equal to the probabilities” (Gillies 2000, p. 161). On the other hand, he is also very clear that single case chances are not propensities, but may only be interpreted as subjective (ibid., pp. 119–120). The interpretation in the text above seems to make most sense of Gillies’ various commitments, but in any case nothing much hinges on this subtlety. Most of the arguments raised in this paper, and in particular those in Sections 4 and 5, go through regardless.
But see Hájek (1997) for a review.
Alternatively, probabilities maybe tested by observed experimental frequencies. Note that the view as described above has affinities with single case propensity theories such as those due to Miller and Fetzer. However, these authors do not really relinquish the identity thesis–they do not separate as cleanly as I do propensities from their probabilistic manifestations.
These terms were introduced in (Suárez 2013) with slightly different meanings but amounting to the same idea that the conjunction of both makes up the full identity thesis. Note that strictly speaking the bi-conditional applies to conditional probabilities only but, as I go on to make clear in Section 5, there is a formal way to render all probabilities explicitly conditional.
Although I am unsure about Popper himself, who sometimes (for instance, in describing how the propensity interpretation ‘takes the mystery out of quantum mechanics’), writes as if he does not countenance any meaningful concept of subjective probability. But this seems rather extreme. Donald Gillies certainly countenances subjective probabilities, as distinct from objective propensities, and defends a kind of pluralism regarding probability (Gillies 2000, Ch. 8). Similarly Carnap (1966), Hacking (1975, 1990), Mellor (2005) all accept at least two different senses of probability, roughly along the lines of the distinction between the objective (relating to physical states) and the subjective (relating to mental or belief states).
Hajek (2003).
Humphreys (1985, p. 561) assumes that emission time is t0, and that it is the physical facts at some strictly later time t1 that fix the propensities. Nothing essential in what follows depends on this assumption, so I shall assume that t1 = t0 without loss of generality.
Strictly speaking it states that its propensity to be transmitted is zero, but I assume throughout that having propensity zero is equivalent to having no propensity.
He writes (Humphreys 1985, p. 561): “[…] The propensity for a particle to impinge upon the mirror is unaffected by whether the particle is transmitted or not”.
One of the most widely discussed such arrangements in the literature is known as Hesslow’s example, where a particular variable is causally related to an effect via two different intermediate routes, one route involving an inhibitor and the other route involving a producer, so finely balanced that no correlation is apparent at all between cause and final effect (Hesslow 1976). The example has famously provided grounds against statistical theories of causation as probability raising in general (for instance, see Cartwright 1989, pp. 99–100).
This may not be Gillies’ considered response, however, since in (Gillies 2000, p. 132), it is asserted that ‘to say that P (A/B&S) = q means that there is a propensity if …’, which shows that Gillies gives a propensitiy interpretation to both fundamental and non fundamental conditional probabilities, insisting only on the non-reversibility of the former.
Gillies, ibid, pp. 161 ff. The exposition in the text differs slightly from the original, on account of our present interest to draw out the consequences of probability systems for the identity thesis.
An ordinary probability space (Ω, F, P) contains a set of possible outcomes of chance trials, or outcome space (Ω), a Borel field F of subsets of the outcome space, and a probability function or measure P defined over the elements of the outcome space.
The Axiom of Independent Repetitions states that if (SS, Ω, F, P) is a probability system, then so is (SS n, Ωn, Fn, P(n)), where SS n is the sequence of repetitions formed out of repeatedly choosing the same n-tuple of elements of SS; Ωn is the n-fold Cartesian product of Ω; Fn is the minimum Borel field of subsets of Ωn that contains F; and the measure P(n) on Fn is the n-fold product measure of the measure P on F. (Gillies, ibid, pp. 164–167).
The account is presented in (Suárez 2013), which also explains the reasons why this particularly non-empiricist conception of propensities may be said to be “pragmatist”.
An alternative interpretation of i)–iii) above, which makes the point even sharper, associates the propensities to dynamical properties of either the set up or the photons themselves as they travel from the source, at different instants of time. Thus propensities are not even described as events–so they cannot even in principle be defined as probabilities.
Similar views had been voiced earlier by Levi (1980, chapter 12).
Or, at the very least, it should be replaced with an equivalent principle stating the appropriate causal relations amongst propensities. Since its expression would involve “»” as opposed to conditional probability, there is no reason to expect any contradiction with the principles i)–iii) above, or with the Kolmogorov axioms.
These are amongst the different intuitions that the expression elicits–thanks to Alan Hajék for pointing out that they are inconsistent with each other, not just with Humphreys’ intuition.
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Acknowledgement
For comments and reactions I thank two helpful EJPS referees, and audiences at the British Society for the Philosophy of Science meeting (2012), at Oxford, Cologne, Lausanne and Bern Universities, and at the London School of Economics. Financial support is acknowledged from the Spanish Government research project FFI2011-29834-C03-01, and the European Commission under the Marie Curie programme grant PIEF-GA-2012-329430.
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Suárez, M. A critique of empiricist propensity theories. Euro Jnl Phil Sci 4, 215–231 (2014). https://doi.org/10.1007/s13194-014-0083-8
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DOI: https://doi.org/10.1007/s13194-014-0083-8