ProbabilisticProbabilistic SquirrelsSquirrels::A Novel Defence of Twofold A Novel Defence of Twofold

CausalityCausality

Federica Russorusso@lofs.ucl.ac.be

Centre de Philosophie des SciencesUCL

OverviewSuppes’ Probabilistic Causality:Main concepts and some criticisms

The Golf Environment:Improbable Consequences, Negative Causes, and Twofold Causality

The Fallacy of Probabilistic Squirrels:A Bayesian Argument

Probabilistic Quantitative Squirrels:A Novel Defence of Twofold Causality

Suppes’ Probabilistic Theory

Barometer Reading, Rain, and Air Pressure

P (R | B) > P (R) The barometer -prima facie- causes rain

P (R | B A) = P (R | A) The barometer is a spurious cause of rain,

P (R | B A) P (R | B) i.e. has no real effect

Is Suppes’ Theory Adequate?

Pregnancy, Thrombosis, and Contraceptive Pills

P (T | C) > P (T) Contraceptive pills cause thrombosis

ButBut

P (T | C) < P (T) Contraceptive pills may prevent thrombosis

SinceSince

P (T | P) > P (T) Pregnancy may also cause thrombosis

AndAndP (P | C) < P (P) Contraceptive pills prevent

pregnancy

The Golf Environment:Improbable Consequences

Clumsy golf players:Hitting a tree-limb makes a birdie even more improbable

If the birdie occurs,relativisation of causal concepts to the causal backgroud is helpful

The Golf Environment:Negative Causes

Nuisance Squirrels:

Squirrels’ kicks lower the probability of birdies

NeverthelessNevertheless

A squirrel’s kick may causecause a birdie

Probabilistic Squirrels save PT, seemingly

Population-level causationP (E | C) < P (E) Squirrels’ kicks are negativenegative causes

Token-level causationP (e | c) > P (e) The squirrel’s kick is a positivepositive cause

Priors: P (E) = .90 ; P (C) = .15 ; P (E C) = .10

Hence: P (E | C) = .66 < P (E) =.90Squirrels are negative causes

Update: Pn (C) = .90

Jeffrey’s RulePn (e) = P (e | h) Pn (h) + P (e | h) Pn ( h)

Pn (E) = .69 < P (E) = .90

The squirrel is still a negativenegative cause!

Bayesian squirrels reveal a fallacy

Suppes’: causal relations among quantitative properties

X, Y, Z Random variables

P (X ≤ x)P (Y ≤ y) Probability distributionsP (Z ≤ z)

cov (X, Y) Measure of association between X and Y

(X, Y) Standardized measure of association between X and Y

Quantitative Barometric Changes

X : number of rainy daysY : height of the barometer readingZ : index of air pressure change

P (X ≤ x | Y ≤ y) > P (X ≤ x) The barometer is a prima facie cause

(X, Y) > 0

P (X ≤ x | Y ≤ y, Z ≤ z) = P (X ≤ x | Z ≤ z) The barometer is a spurious cause

(X, Y | Z = z) = 0

Quantitative Probabilistic Squirrels

1 if the ball falls into the cupCup X

0 if the ball does not

1 if the squirrel kicks the ballSquirrel Y

0 if the squirrel does not

Quantitative Squirrels are negative causes

P (X = 1 | Y = 1) = .66 < P (X = 1 | Y = 0) = .94

Squirrel –Y-

Cup –X-

1 0 Tot1 .10 .80 .900 .05 .05 .10

Tot .15 .85 1

The probability of a birdie given the squirrel’s kick does exist

80%

10% 5% 5%

Cup givennot Squirrel

Cup givenSquirrel

Not Cupgiven notSquirrel

Not Cupgiven

Squirrel

Still, do you want the squirrel to be a token positivepositive cause?

Token probability trajectories face two problems:

the exact specification of the causal context and of all the factors involved

the reference to different hipothetical corresponding type-populations

Sketch of a solution

Do we really need a fully developed token probabilistic theory?Maybe no …

Is a Bayesian framework promising?Maybe yes …

To sum up …The probabilistic theory of causality is sound, consistent, and promising

Twofold causality is defensible

Probabilistic squirrels are acquitted,provided that we adopt:

a twofold conception of causality and a Bayesian framework