Fetzer’s Solution tothe Reference Class Problem

Causality, Probability, and Counterfactuals

Lorenzo Casini

Philosophy, Kent

<L.Casini@kent.ac.uk>

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Outline

• Two questions (What is RCP)• Two problems (Why does RCP really matters)• Fetzer’s solution of MRCP• A possible defence of Fetzer’s proposal• Applying Fetzer’s solution to CBNs• Summary

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Two Problems:

• MRCP: existence of a relevant reference class• ERCP: justifying a probability value as the right one

I will focus on:solution of MRCP, which is

• Ontologically acceptable (reference class description points to something out-there-in-the-world)

• Epistemologically acceptable (relevant reference class is knowable in principle)

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Two Questions:

• What is p that x is A given that x is a ‘such-and-such’ event?Many Rs admissible many answers:

• What is p that x is A because x is a ‘such-and-such’ event?( and related counterfactual)Unique answer requires unique R

Fetzer: MRCP is about single-case true statistical explanations

,...2211 p),P(A|Rp)P(A|R

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Fetzer’s (and Pearl’s) recipe:

• Explanatory relevance is nomic or causal relevance (‘this x is A because it is R’ if ‘being R c being A’)

• R is a complete set of causally relevant factors

• Factors are objective (mind independent)

• S-R is only a guide to C-R

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Fetzer’s solution to MRCP

Probability is the disposition of a set-up to:• give an outcome on a single trial• generate long-run frequencies for the possible

outcomes

Such disposition is possessed by every member of a maximally specific reference class R

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OntologyThe classifying of x as member of R at t entails necessarily:

• x’s possession of a permanent dispositional property χ at t

Syntax (i) (subjunctive conditional) (ii) (causal conditional)

Semantics of of worlds where is true over worlds

where is true is n

kF

)χ)()(( xtRxttx *)]()[)(( xtOxtTRxttx i

ni

*1atO i

11. atTRat iklim

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HypothesisingF is C-R if

Testing• Causal conditionals are corroborated if frequency

distribution approximates normal distribution (Bernoulli’s theorem)

• F is C-R if, for :

Notice: Testing requires randomness assumption(i.e.: K’s maximal specificity)

*)..)()(*).(..)()(( xtOFxtxtTRxttxxtOFxtxtTRxttx in

iim

i

).|().|( FxtRxtOxtFFxtRxtOxtF nn

nm

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Possible defence of Fetzer’s proposal• Existence:O. Either there are many reference classes at different t or

the relevant one is the closest to the outcome which occurs with p=1 (Hanna, 1982)

P. Completeness assumption refers to a set-up type at under assumption that nothing interferes

• Explanation:O. Completeness is too demanding—prevents any

explanation to be true (Humphreys, 1982)P. F.’s proposal is suitable for HPs on repeatable and

controlled conditions

0t

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• Knowability:

O. type-level probabilities say nothing about single-case propensities (Pollock, 1990)

P. F.’s proposal is epistemologically acceptable if evidence can confirm completeness of R

• Testability:

O. Completeness is untestable in principle (Gillies, 2002)

P. For frequencies to confirm completeness, we must assume underlying conditions (causal structure & dispositional strength) are stable

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In sum:

assume• randomness of outcomes• stability of causal structure & dispositional strength• sequences are long enough

(Tests are reliable)

then,• we can distinguish causally relevant properties among

prima facie causally relevant ones• if R is objectively homogeneous, no other statistically

relevant property will be causally relevant

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Applying Fetzer’s solution to CBNs

Causal graph on • Any is probabilistically independent, given , of

any set of variables not containing its descendents (CMC)

• Pearl: the relevant R is the sum total of and (X is deterministically dependent on its known and unknown causes)

• But: what do correspond to? how to know whether CMC holds for ?

• Plus: no genuinely statistical explanations!

• Do we need determinism to solve MRCP?

nXXXV ,...,, 21

iPa

iX ipa

ie

ieie

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Under Fetzer’s dispositional interpretation

is single-case, objective, testable

• F.’s solution involves time reference (applicable to dynamic BNs?)

• Only are objective (mind independent)( play the role of ceteris-paribus clauses: ‘nothing interferes’)

• Counterfactual analysis needs no deterministic assumptions to solve MRCP: e.g.: ‘R’=‘L.F’, x belongs to R and brings about O with probability m. Had x belonged to , would O have occurred with probability n, where ?

*)( utxutTutpa ini

i

iPaie

FL .nm

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Summary

MRCP is about • the existence of a complete reference class description

which explains the probabilistic possession by an individual of a certain attribute via a statistical dispositionSingle-case dispositional claims

• have their truth conditions in (intensional) limiting frequencies over possible-worlds. When these are true (/corroborated), we know (/have good reasons to believe in):

• completeness of reference class (thereby solving MRCP—without determinism), and

• the ontology of single-case dispositions (which give causal conditionals the desired explanatory power)

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Open issues

Still, when it comes to testing…

remain

F.’s solution does not rely on (extensional) limiting frequencies, yet on other (not less) controversial assumptions. To what extent can we

• rely on (in/)stability of causal structure & dispositional strength to (dis/)confirm completeness?

• extend propensity claims corroborated within a population (where unknown causes might be ‘friendly’) to other populations/individuals?

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Thank you

What is What is

your Reference

Class, darling? reference

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References

Cartwright, N. (2003), “What makes a capacity a disposition?” Causality: Metaphysics and Methods discussion paper 10/03, London School of Economics Centre for Philosophy of Natural and Social Science

Dawid, A. P. (2000a), “Causal Inference Without Counterfactuals”, Journal of the American Statistical Association, 95: 407-424

Dawid, A. P. (2000b), “Causal Inference Without Counterfactuals: Rejoinder”, Journal of the American Statistical Association, 95: 444-448

Fetzer, J. H. (1970), “Dispositional Probabilities”, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 473-482

Fetzer, J. H. (1981), Scientific Knowledge. Causation, Explanation, and Corroboration, Boston Studies in the Philosophy of Science, D. Reidel Publishing Company, Dordrecht, Vol. 69

Fetzer, J. H. (1982), “Probabilistic Explanations”, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 2: 194-207

Gillies, D. (2000), “Varieties of Propensities”, British Journal for the Philosophy of Science, 51: 807-835Gillies, D. (2002), “Causality, Propensity, and Bayesian Networks”, Synthese, 132: 63-88Hájek, A. (2007), “The Reference Class Problem is Your Problem Too”, Synthese, 156: 563-585Halpern, J. Y., Pearl, J. (2005), “Causes and Explanations: A Structural-Model Approach. Part II:

Explanations”, British Journal for the Philosophy of Science, 56: 843-887Hanna, J. F. (1982), “Probabilistic Explanation and Probabilistic Causality”, PSA: Proceedings of the

Biennial Meeting of the Philosophy of Science Association, 2: 181-193Humphreys, P. (1982), “Aleatory Explanations Expanded”, PSA: Proceedings of the Biennial Meeting of

the Philosophy of Science Association, 2: 208-223Pearl, J. (2000a), Causality. Models, Reasoning, and Inference, Cambridge University PressPearl, J. (2000b), “Causal Inference Without Counterfactuals: Comment”, Journal of the American

Statistical Association, 95: 428-431Pollock, J. L. (1990), Nomic Probability and the Foundations of Induction, Oxford University PressReichenbach, H. (1949), The Theory of Probability, University of California PressWilliamson, J. (2006), “Dispositional versus Epistemic Causality”, Minds and Machines, 16: 259-276Williamson, J. (2007), “Causality”, in Gabbay, D. and Guenthner, F. (eds.): Handbook of Philosophical

Logic, Springer, 14: 89-120