Abstract
This introduction summarises the main themes of the book, and sketches a theory of the relation between propensities and probabilities that is consistent with many of the claims stated throughout the book. The theory is developed around four different theses concerning respectively: the objective nature of probability in physics, the key role played by transition probabilities in modelling statistical phenomena, the apparent redundancy of philosophical interpretations of probability, and last but not least, the ubiquity of causal concepts and presuppositions underlying probability models.
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- 1.
Bacciagaluppi’s terminology employs the technical notion of an n-fold joint distribution, which is standard in the literature on stochastic processes (see e.g. Doob, 1953). According to this terminology, states 1 to n appear in the subscript of the probability function, and time indexes in its variable range. We then consider the n-fold joint probability distributions that the n states define over the time indexes. This terminology is more convenient for the derivation of technical results but it strikes me as less intuitive, at least for the purposes of this chapter.
- 2.
These notions are again expressed in my own terminology. The notation of n-fold distributions has, undoubtedly, an advantage at this point since it allows us to distinguish the concept of symmetry of the transition probability from the concept of detailed balance (see Bacciagaluppi’s Section 3, where it is also claimed that under standard conditions these concepts are equivalent as statements of time-symmetry). But the distinction plays no role in this introductory essay which focuses instead on conceptual issues regarding objective probability.
- 3.
So, importantly, a backwards transition probability is not the forwards transition probability of the time-inverse of the state change: \({\textrm{Prob}}_{j/j + 1} (S(t_{j + 1}) / S(t_j)) \ne {\textrm{Prob}}_{j + 1/j} (S(t_j )/S(t_{j + 1} ))\), with \(t_{j + 1} > t_j\). The latter is rather a different transition probability altogether, belonging to an entirely different Markov process.
- 4.
- 5.
Van Fraassen (1989, 303–304).
- 6.
See Strevens (1998, 231) for further discussion.
- 7.
- 8.
- 9.
Gillies (2000a, 47–49), where several examples from physics are provided, such as the viscosity of gases and Bose Einstein statistics.
- 10.
See Gillies (2000a, 147).
- 11.
A test statistic for an experiment is a random variable X, whose value can be calculated as a function of the data sampled, \(X \{ e_1 ,e_2 ,e_3 , \ldots ,e_n \}\), and that can be taken to represent the outcome of the experiment. Note that the same experiment may yield different values for the test statistic, depending on the data sampled.
- 12.
Howson and Urbach (1993, 210–212). In their example we may choose either to terminate the experiment as soon as 6 heads occur, or rather after 20 trials regardless of the outcome. The size of the outcome space is then predetermined in the latter case (= 220) but not so in the former. Even if the outcome spaces happened to have the same size in both cases (because say the 6th head happens to occur on the 20th trial), it would still be the case that the stopping rule could affect the result of the application of the falsifying rule, falsifying it in the former but not the latter case.
- 13.
Note that Gillies disagrees that a falsificationist methodology is in any way threatened by Howson and Urbach’s argument. See particularly the discussion in his interesting review of their book (Gillies, 1990, 90–97). Howson and Urbach respond in the 2nd edition of their book (214–215). This debate turns on whether or not the stopping rule is relevant to the performance of the experiment, and therefore relevant to the evaluation of the application of the falsifying rule. It is surprising that this debate does not yet appear to have been linked to the question of the nature of the probabilities involved, and in particular whether they are subjective or objective probabilities.
- 14.
Should there be one? The presumption that there should is of course tantamount to the view that thermodynamics should be in some sense reduced to statistical mechanics. It is controversial whether such attempts have been successful. Moreover it is unclear that they should be in order to ground thermodynamic irreversibility. See for instance Sklar (1993, Chapter 9). Such interesting questions are beyond the purview of this essay or this book.
- 15.
A measure usually defined over the semi-closed intervals of the real line (see Halmos, 1974, 65ff.)
- 16.
Sklar (1993, 159–160).
- 17.
For a thorough critique see Earman and Rédei (1996).
- 18.
See Tumulka (2007) for the distinction and a development of the ‘flash’ ontology.
- 19.
In the case of the famous ‘bilking’ argument (Black, 1956), the assumption is simply that an event c is the positive cause of an event e that lies in its past. The issue is then how to prevent the bilking of c after e has occurred. For if we prevent c from happening after e has already occurred, then this would generate the inconsistency that both ‘c is the cause of e’ and ‘c is not the cause of e’ are simultaneously true. Much will depend on whether ‘bilking’ is actually physically possible in the particular circumstances that give rise to c and e. Similarly for the type of inconsistency that causal loops may generate: much will hinge on the particular circumstances that bring about the EPR correlations.
- 20.
‘Hidden autonomy’ is Van Fraassen’s (1982) terminology.
- 21.
But does statistical dependency reflect causal dependency? Arguably the relationship is more complex and subtle. First, it is well known that statistical dependencies may mask hidden factors or hidden common causes. And second, the relation of conditional probability P(x/y) need not indicate that the conditioned upon event y is a direct cause of the event x. This requires a further assumption (see Section 1.6 in this essay). I will follow Berkovitz here and assume for the sake of argument that causal dependencies can be read off statistical relations. In the second part of this chapter, I argue that conditional probabilities are not generally a reasonable way to read propensities.
- 22.
Throughout his paper Berkovitz assumes a single-case propensity interpretation of probabilities. But he shows that analogous results stand if the probabilities are understood as frequencies.
- 23.
The name ‘Budapest school’ was introduced by Jeremy Butterfield (2007, 807).
- 24.
For the distinction between the ‘criterion’ and the ‘postulate’ of common cause see Suárez (2007b).
- 25.
See Gyenis and Rédei’s Definition 3.1.
- 26.
See Gyenis and Rédei’s definition 4.1. A common cause variable C K has size 2 if it has two values. For instance an indicator function (on-off) can be represented as a size two variable (C, ¬C).
- 27.
Gyenis and Rédei leave open what this further conditions may be, which seems wise since their aim is to describe formal models applicable to any physical set ups. In causal modelling one would of course like to know more about this relation, and in particular the physical conditions that must obtain for A, B to be causally independent in the prescribed sense.
- 28.
- 29.
Mellor (1971).
- 30.
I introduce irreducible dispositions into Bohmian mechanics in Suárez (2007a, Section 7.2). However, I was not the first person to suggest such a reading. Pagonis and Clifton (1995) are an antecedent (although to my mind they mistakenly understand dispositions relationally, and identify them with aspects of Bohmian contextuality). An attempt closer to my own ideas is due to Martin Thomson-Jones (Thomson-Jones, unpublished). We both defend irreducible dispositions with probabilistic manifestations for Bohmian mechanics but unlike Thomson-Jones I restrict the applicability claim to the causal or maximal interpretation. Thomson-Jones’ unpublished manuscript is dated after the submission date of the final version of my paper. However, I was in the audience both in Bristol (2000) and Barcelona (2003) where preliminary versions of Thomson-Jones’ paper were presented. Although I don’t recall the details of these talks I am sure I was influenced by them, as well as many friendly chats with Martin over the years – for which I am very grateful.
- 31.
It is not surprising that such theories have already received interpretations in terms of dispositions – see Frigg and Hoefer (2007) and Suárez (2007a, Section 7.1).
- 32.
- 33.
- 34.
David Lewis (1997, 149 ff.) introduced the idea of causal bases for dispositions. Bird (2010) discusses objections to the idea that stimulus conditions cause dispositions to manifest themselves. For the purposes of this introduction I have ignored stimuli and concentrated on the disposition – manifestation relation itself (e.g. in the discussion in Sections 5–6).
- 35.
On the assumption of a fixed past and an open future (CP j+1) does not express anything informative since \(P_{j + 1} (S(t_j )) = 1\) and \(P_{j + 1} (S(t_{j + 1} )/S(t_j )) = P_{j + 1} (S(t_{j + 1} ))\) for any states \(S(t_j )\), \(S(t_{j + 1} )\). But Bacciagaluppi is interested in the meaning that these expressions, and the corresponding concepts, may have in the absence of any assumptions regarding becoming or any other asymmetry in time. So he is right in considering them as distinct possibilities. The only reason I ignore (CP j+1) in what follows is that all the considerations in the text above against reading (CP j ) as a transition probability apply just as well to it.
- 36.
A different further question is whether these probabilities (in particular TP and CP j , whenever they are both well defined) should coincide numerically for the initial and final states of any state transition. A study of the conditions under which they coincide is beyond the reach of this essay – but it seems to me to be an interesting and promising research project.
- 37.
- 38.
See Arntzenius (1995, esp. Section 2) for a detailed example and discussion.
- 39.
Penrose (1989, 355–359) defends an apparently similar view regarding the quantum mechanical algorithm for computing transition probabilities (the Born rule) in general. He claims that the algorithm can err if applied to compute backwards state-transitions: ‘The rules […] cannot be used for such reversed-time questions’ (ibid, p. 359). The representation of transition probabilities proposed here makes it clear why this should be the case.
- 40.
The view of propensities that I shall be defending here is very much my own (see Suárez, 2004, 2007a), and none of the contributors in the book has explicitly committed to it. However I believe that this view, or a similar one, is required for the coherence of many pronouncements made in the book, particularly in the third part. If so, we may take this or a similar view to be implicit in the book, and its defence in this section to provide support for it.
- 41.
- 42.
Long run propensities as tendencies to generate infinite sequences seems to be what the early Popper defended in his classic (1959), and as tendencies to generate long but finite sequences by Gillies (2000a, Chapter 7). Single case propensities are defended by Fetzer (1981, Chapter 5) and Miller (1994).
- 43.
- 44.
The different interpretations are then classified as follows: Fetzer (1981) defends a single case repeated conditions interpretation, while Miller (1994) defends a single case state of the universe interpretation. Gillies (2000a, 130–136) argues that these interpretations succumb to Humphrey’s paradox, and defends instead a long run repeated conditions interpretation.
- 45.
Humphreys (2004).
- 46.
Humphreys actually lists a fourth case, the causal interpretation (Humphreys, 2004, 673). However, the causal interpretation is not really on a par with the other three since it is not per se a dynamical interpretation of the evolution of propensities. In fact it does not seem to exclude any of the other three dynamical interpretations, being rather compatible with any of them.
- 47.
- 48.
For the convenience of the story, I am assuming that the relata of causation are facts along the lines of Mellor (1995). But the argument does not hinge on this assumption.
- 49.
This need not rule out absolute propensities, although some commentators – notably Gillies (2000a, 131–132) – go further and claim that all propensities are implicitly if not explicitly conditional. In this view a propensity interpretation of probability is always of (and only of) conditional probability.
- 50.
Humphreys (1985, 561).
- 51.
I have adopted Humphreys’ suggested terminology and refer to propensities as Pr(–) and probability functions as either Prob(–) or simply P(−).
- 52.
Humphreys (1985, 561; 2004, 669).
- 53.
Humphreys (1985, 562).
- 54.
McCurdy (1996).
- 55.
See Humphreys (2004). My objections below to CI are very different in nature and cannot be answered by means of new examples.
- 56.
Milne (1986).
- 57.
See Maudlin (1995), particularly Chapter 5.
- 58.
In fact many of the arguments against backwards in time causation turn out to depend on the fine grained space-time structure of the putatively refuting examples. Others, such as the bilking argument, attend to agency only, but seem inconclusive. See Black (1956) and Dummett (1964) for two classic sources and discussion.
- 59.
See for instance the table in Humphreys (2004, 677).
- 60.
We may wonder about the status of conditional independence in other interpretations of propensities. CI holds in the temporal evolution interpretation – since the propensity of I t2 is updated at time t 2. So \(\Pr _{t2} (I_{t2} /T_{t3} B_{t1} ) = \Pr _{t2} (I_{t2} /\neg \;T_{t3} B_{t1} ) = \Pr _{t2} (I_{t2} /B_{t1} ) = 1{\textrm{ or }}0\). But it fails in the renormalisation interpretation since \(\Pr _{t3} (I_{t2} /T_{t3} B_{t1} ) \ne \Pr _{t3} (I_{t2} /\neg T_{t3} B_{t1} )\) in general. However, Humphreys (2004, 673) finds that a similar principle holds in the renormalisation interpretation, namely the fixity principle. (The fixity principle states that: \(\Pr _{t1} (I_{t2} /T_{t3} ) = 0{\textrm{ or }}1\), which holds in the renormalisation interpretation since \(\Pr _{t3} (I_{t2} /T_{t3} ) = 0{\textrm{ or }}1\)). In all cases, I contend, Assumption 2 is implicit in the derivation of CI.
- 61.
Humphreys (2004, 675)
- 62.
As good as they come – typically not up to uniqueness. In particular, and rather to the point, the fourth Kolmogorov axiom is sometimes disputed – see, e.g. Hajek (2004).
- 63.
Why suppose that objective probability, or chance, requires any interpretation at all? After all many theoretical concepts bring their own interpretation and/or require no interpretation. Elliott Sober for one has recently argued for a no-theory theory of probability in Sober (2005).
- 64.
I therefore assume that the actualisation of a propensity is tantamount to a state transition from the propensity to the manifestation property. This is necessarily the case whenever the new manifestation property is incompatible with the original propensity. Otherwise it is a contingent matter of fact whether the actualisation process entails a transition, but this seems plausible in most ordinary cases. Thus the smithereens of a broken glass are rarely themselves fragile. And even if they were, the property of fragility would no longer be a property of the original entity. So it is arguable that the evolution of the system as described is best represented by means of a state transition anyway.
- 65.
For the distinction between ‘productive’ and ‘dependence’ or counterfactual causality, see Hall (2004).
- 66.
- 67.
The problem is most acute for long run propensity theories. Gillies (2000a) attempts to solve the problem by appealing to the notion of a falsifying rule for hypotheses. But this is a controversial solution as discussed in Section 1.2.2.
- 68.
There is an interesting question here for the ‘causal’ notation alternative mentioned earlier. In that case we would write \(P(F \hookrightarrow A_i ) = p_i\) with \(\Sigma p_i = 1\), and \(p_i = p_j\) for any i, j. Here the application of the principle of indifference would lead us to infer objective facts. However, these facts do not regard the distribution of propensities but refer exclusively to the causal efficacy of propensities in generating distributions. It is an open question to what extent such an inference is prohibited by the sort of arguments routinely employed against the principle of indifference. Bertrand style paradoxes, for instance, are prima facie inapplicable given the apparent absence of any causal relations in those geometrical examples. This is an interesting topic for further work.
- 69.
Note that failure of symmetry is the case in the ‘causal’ notation too. Thus it does not follow from \(P_c (F\hookrightarrow B)\) that \(P_c (B\hookrightarrow F)\). It does not, in fact, follow that F has any causes at all, never mind that B is one of them.
- 70.
The same conclusion follows in accordance to the ‘causal’ notation. Humphreys conditions would be formalised as follows: \({\textrm{(i) }}P_{t1} (I_{t2} B_{t1} \hookrightarrow T_{t3} ) = {\textrm{p}};{\textrm{(ii) }} P_{t1}(B_{t1}\hookrightarrow I_{t2} ) = q, \textrm{where}\,\, 1 > q > 0; {\textrm{(iii) }}P_{t1} (\neg I_{t2} B_{t1} \hookrightarrow T_{t3} ) = 0\). Since Bayes Theorem has no application, no contradiction can ensue.
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Suárez, M. (2011). Four Theses on Probabilities, Causes, Propensities. In: Suárez, M. (eds) Probabilities, Causes and Propensities in Physics. Synthese Library, vol 347. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9904-5_1
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