Groundwork for a Probabilistic Analysis of Causation: Foundational Issues in the Philosophy of Probability [email protected].

Groundwork for a Probabilistic Analysis of Causation:

Foundational Issues in the Philosophy of Probability

[email protected]

‘Orthodox’ Probabilistic Analyses of Causation

• Probabilistic analyses of causation typically state that event c is a cause of distinct event e if (and only if) c raises the probability of e relative to background conditions b.

• Various ways of cashing out this notion of probability-raising.

• Orthodox analyses (Good 1961a, b; 1962; Reichenbach 1971; Suppes 1970; and Kvart 2004) cash it out in terms of an inequality between conditional probabilities:

)( ) .|~() .|( PIBCEPBCEP

Groundwork for a Probabilistic Analysis of Causation 2

Lewis’s Objection to Orthodox Analyses

“Conditional probabilities, as standardly understood, are quotients. They go undefined if the denominator is zero. If we want to say, using conditional probabilities, that c raises the probability of e, we will need probabilities conditional on the non-occurrence of c (plus background, perhaps). But there is no guarantee that this conditional probability will be defined. What if the probability that c occurs (given background) is one? … For that matter, what if we want to apply our probabilistic analysis of causation to a deterministic world in which all probabilities (at all times) are extreme: one for all events that do occur, zero for all that don’t? The requisite conditional probabilities will go undefined, and the theory will fall silent.” (Lewis, 1986, p.178)



• Kolmogorov (1933) axiomatization defines conditional probability as a ratio of unconditional probabilities:

• P(Y|X) undefined where P(X) = 0.

)( 0)( ,)(

).()|( RATIOXP

XP

YXPXYP



• But suppose, for some cause c, P(C) = 1. Then P(~C) = 0 and P(~C.B) = 0.

• The term on the RHS of (PI) will be undefined and so orthodox analyses fail to yield the result that c is a cause of e.

• But surely probability 1 events can act as causes!


Lewis’s Solution

• Lewis’s solution is to adopt a non-orthodox probabilistic analysis on which the relevant notion of probability-raising not in terms of conditional probabilities, but in terms of counterfactual conditionals.

• But there are worries about whether a semantics for the relevant counterfactual conditionals can be given that will yield the right results about causation when combined with an appropriate non-orthodox probabilistic analysis.

• We needn’t face these worries because the problem of probability 1 causes for the orthodox approach can be diffused.


Probability 1 Events

• Probability 1 events needn’t be determined to happen (i.e. needn’t be necessary given the laws and circumstances).

• In solving the problem of probability 1 causes for orthodox analyses it is important to distinguish between those probability 1 causes that are determined and those that are not.

• The solution to the problem is different in the two cases.


Probability 1 Causes that are Not Determined

• d = a perfectly reliable device that will randomly select a point on the Earth’s surface.

• Bookmaker offers Warren a bet on the result of the next random selection:

Warren pays bookmaker

Bookmaker pays Warren

Device selects point on equator

$10 billion $0

Device fails to select point on equator

$0 $1



• Equator has zero area and so the probability of an equatorial point being selected is 0. The probability of a non-equatorial point being selected is 1 (though this is not determined to happen). The expected payoff of the bet to Warren is $1.

• Warren accepts the bet, d is put into operation and selects a non-equatorial point. Warren wins $1.

• d’s selection of a non-equatorial point caused Warren to win $1. But, if (RATIO) is correct, the probability of Warren’s winning $1 conditional upon d’s not selecting a non-equatorial point (a probability 0 event) is undefined.

• Orthodox analyses fail to yield the correct result.



• The problem just described depends upon the correctness of (RATIO). But (RATIO) is false, as Hájek (2003) has shown.

• Hájek argues that there are well-defined conditional probabilities with probability 0 conditions. For example, although P(Equator) = 0 the following equalities all hold:

P(Equator|Equator) = 1

P(WesternHemisphere|Equator) = ½

P(Brazil|Equator) ≈ 0.06

• These probability values are generated by facts about the probabilistic set-up. Yet (RATIO) leaves all of these values undefined.



• These and other considerations – in particular the apparent relativity of probabilities to reference classes and probabilistic set-ups – lead Hájek to argue that Kolmogorov’s axiomatization should be rejected in favour of one (e.g. Popper, 1972) which takes conditional probabilities as primitive and unconditional probabilities as derivative.

• If conditional probabilities are primitive, then there is no pressure to deny the existence of the probabilities required by the orthodox analysis.


Probability 1 Causes that Are Determined

• But whilst one might countenance well-defined probabilities conditional upon probability 0 events (such as the selection of an equatorial point), one might still reasonably doubt that there are well-defined probabilities conditional upon events that both have probability 0 and are physically impossible.

• This is a problem for orthodox analyses because, in a deterministic world, the absence of any given cause both has probability 0 and is determined not to happen.

• But surely we want to allow that causation and determinism are compatible!



• But deterministic causation doesn’t pose any genuine problem for orthodox analyses.

• There are events with non-trivial probabilities (probabilities other than 1 or 0) even in deterministic worlds.

• There are therefore well-defined probabilities conditional upon the absence of these events (indeed this would be so even if (RATIO) were correct).

• Moreover, these non-trivial probabilities are not of the irrelevant, epistemic, sort. They are objective chances.



1. Many of the high-level or special sciences give probabilistic laws for the events falling under their purview.

2. These laws are genuine, objective laws. They fulfil all the usual criteria by supporting counterfactuals, being confirmed by their instances and playing the right role in explanation and prediction.

3. Together with the initial history of the world, these laws entail non-trivial probabilities for the events that they cover.

4. Even if the world turned out to be deterministic, these laws would remain probabilistic.

5. So there are lawfully projected, non-trivial probabilities even in deterministic worlds.



• But Schaffer (2007) has argued that the non-trivial probabilities projected by the special scientific laws of deterministic worlds cannot be genuine, objective chances because they don’t play the role of chance.

• In particular, Schaffer argues that such non-trivial deterministic probabilities don’t connect in the correct way to rational credence and possibility (ibid., p.132).

• I shall focus on the former connection, but worries about the latter connection can be diffused in a similar fashion.



• Lewis’s (1986) Reformulated Principal Principle:

C = a reasonable initial credence functiont = a timew = a worldChtw = the chance function

p = a proposition within the domain of the chance functionHtw = the proposition giving the total history of w through t

Lw – the proposition giving all of the laws of w

)( )().|( RPPpChLHpC twwtw

16Groundwork for a Probabilistic Analysis of Causation


• Schaffer argues that non-trivial deterministic chances are incompatible with the RPP:

1. Suppose pe is the proposition that some event e occurs and Chtw(pe) is a non-trivial deterministic chance: 1 > Chtw(pe) > 0.

2. Since w is deterministic, Lw.Htw (t>0) entails either pe or ~pe.

3. (From 2) C(pe|Htw.Lw) equals either 1 or 0.

4. (From 3 and RPP) Chtw(pe) equals either 1 or 0.

5. (From 1 and 4) 1 > Chtw(pe) > 0 and Chtw(pe) equals either 1 or 0. Contradiction! (Or so it seems ...)



• Schaffer has apparently shown that the supposition that the probabilistic special scientific laws of deterministic worlds may project a non-trivial chance for pe contradicts the RPP which delivers a trivial chance for pe.

• But there is only an inconsistency in allowing two or more divergent chances to attach to the same proposition in the same world at the same time if chance is a function of just three arguments: <p, w, t>.

• If chance is a function of four arguments, there is no contradiction.



• What is the fourth argument?

• The existence of laws of different levels puts pressure on us to accept divergent chances, precisely because these laws entail divergent chances. So chances are level-relative.

• Chtwl(p) = ‘the l-level chance of p at t and w’.

• Chtwl(p) is just the chance entailed for p by the l-level laws of w when taken together with the history of w through t.

• If chances are level-relative then RPP is not a valid reformulation of the original PP. Relative to the chances of a level, only the laws of that level are admissible.



• A valid reformulation is (RPP*):

• Where l = the fundamental level of w, and w = a deterministic world, the chance entailed by Htw.Lwl will be trivial and reasonable credence conditional upon just Htw.Lwl will be correspondingly trivial.

• But where l = a non-fundamental level, the chance entailed by Htw.Lwl need not be trivial and reasonable credence conditional upon Htw.Lwl will be correspondingly non-trivial.

• Non-trivial deterministic chances are therefore compatible with this valid reformulation of the PP, and so they are compatible with the connection from chance to rational credence captured by the PP.

*)( )()|( RPPpChLHpC twlwltw

20Groundwork for a Probabilistic Analysis of Causation


• So even in deterministic worlds there are non-trivial probabilities that are lawfully projected and that guide rational credence. These probabilities therefore play the objective chance role, and should be regarded as objective chances.

• Where c is some event occurring in deterministic world w that is assigned a non-trivial chance by the probabilistic laws of some non-fundamental level l*, there is no problem with a well defined l*-level chance conditional upon the absence of c (plus background b): Chtwl*(E|~C.B).

• There is therefore room for giving an orthodox probabilistic analysis of deterministic causation by cashing out the notion of probability-raising in terms of an inequality between conditional chances of such a level:


)*( ).|~().|( ** PIBCEChBCECh twltwl

Conclusion

• If one rejects (RATIO) and instead takes conditional probability as primitive, then undetermined probability 1 causes do not pose a problem for orthodox probabilistic analyses.

• Recognition of the level-relativity of chances helps dissolve objections to non-trivial deterministic chances. The existence of such chances allows orthodox probabilistic analyses to give the right results in deterministic worlds even if one accepts (RATIO).

• Thus, recognition of the various dimensions of the relativity of probability clears the ground for the construction of an orthodox probabilistic analysis of causation.


Bibliography - I

• Good, I. J. (1961a): ‘A Causal Calculus (I)’, The British Journal for the Philosophy of Science, Vol. 11, No. 44. (Feb), pp.305-318.

• Good, I. J. (1961b): ‘A Causal Calculus (II)’, The British Journal for the Philosophy of Science, Vol. 12, No. 45. (May), pp.43-51.

• Good, I.J. (1962): ‘Errata and Corrigenda’, The British Journal for the Philosophy of Science, Vol. 13, No. 49 (May), p.88.

• Hájek, Alan (2003a): ‘Conditional Probability Is the Very Guide of Life’ in Henry Kyburg Jr. and Mariam Thalos (eds.) Probability is the Very Guide of Life: The Philosophical Uses of Chance (Chicago: Open Court, 2003), pp.183-203.

• Hájek, Alan (2003b) ‘What Conditional Probability Could Not Be’, Synthese, Vol. 137, No. 3 (December), pp.273-323.

• Hájek, Alan (2007) ‘The Reference Class Problem is Your Problem Too’, Synthese, Vol.156, No.3 (June), pp.563-585.


Bibliography - II

• Kolmogorov, A.N. (1933) Grundbegriffe der Wahrscheinlichkeitrechnung, Ergebnisse Der Mathamatk (Berlin: J. Springer); translated (1950) as Foundations of the Theory of Probability, Morrison, Nathan (ed.). (New York: Chelsea Publishing Company).

• Kvart, Igal (2004): ‘Causation: Probabilistic and Counterfactual Analyses’ in Collins, Hall and Paul (eds.) (2004). pp.359-386.

• Lewis, David (1986a): Philosophical Papers, Vol. II (Oxford: OUP)

• Lewis, David (1986b): ‘A Subjectivist’s Guide to Objective Chance’ in Lewis (1986a), pp.83-113.

• Lewis, David (1986c): ‘Postscripts to “Causation”’ in his (1986a), pp.172-213.

• Hoefer, Carl (2007): ‘The Third Way on Objective Probability: A Sceptic’s Guide to Objective Chance’, Mind, Vol. 116, No. 463 (July), pp.549-596


Bibliography - III

• Lewis, David (1994): ‘Humean Supervenience Debugged’, Mind, Vol. 103, No. 412 (Oct.), pp.473-490.

• Popper, Karl (1972): The Logic of Scientific Discovery (London: Hutchinson). (First published 1959).

• Reichenbach, Hans (1971): The Direction of Time (Berkeley and Los Angeles, California: University of California Press). (first published by University of California Press in 1956).

• Schaffer, Jonathan (2007) ‘Deterministic Chance?’, The British Journal for the Philosophy of Science, Vol. 58, pp.113-140.

• Suppes, Patrick (1970): A Probabilistic Theory of Causality, Acta Philosophica Fennica XXIV (Amsterdam: North-Holland Publishing Company).


Groundwork for a Probabilistic Analysis of Causation: Foundational Issues in the Philosophy of Probability [email protected].

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