Online notes that may be helpful

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Lecture 13
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On-line notes that may be helpful

• http//brian.weatherson.org/424/DTBook.pdf
• Probability, conditional probability, objective
  probability, truth tables, decision-making

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A Deterministic Model of Causal Connections

• Fig 7.1
• Consider systems in residual state S.
• Suppose that whenever C is present, it produces
  E.
• And whenever Not-C is present, it produces Not-E.

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Example

• S Normal human body
• C Getting rabies
• E Foaming at the mouth and rapid death
• C is a positive causal factor for E because C
  produces E and Not-C produces Not-E.
• (Assuming nothing else causes foaming at the
  mouth and rapid death.)

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Characterization of Causal Factors

• C is a positive causal factor for E in an
  individual, I, characterized by residual state,
  S, if in I, C produces E and Not-C produces
  Not-E.
• C is a negative causal factor for E in an
  individual, I, characterized by residual state,
  S, if in I, C produces Not-E and Not-C produces E.

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Characterization of Causal Relevance

• If C is either a positive or negative causal
  factor for E in I, with S, then C is causally
  relevant for E in I.
• If C is neither a positive or negative causal
  factor for E in I, with S, then C is causally
  irrelevant for E in I.

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Feature 1 of Model Hypothetical claim

• To say that C is a causal factor for E isnt just
  to say that C and E are both present.
• It also says that if C werent present, E
  wouldnt be.
• Similarly, to say that C is a causal factor for E
  isnt just to say that neither C and E are
  present.
• It also says that if C were present, E would be.
• The logical positivists would be foaming at the
  mouth.

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Feature 2 of Model Residual state is crucial

• The effects of the various factors depend on the
  residual state, S.
• Example Suppose the person gets the rabies
  vaccine. Then contracting rabies will not lead to
  foaming at the mouth and death.

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Feature of Model 3 Determinism

• A system is deterministic between time 1 and a
  later time 2 if its state at time 1 completely
  determines its state at time 2.
• If we rewound time and let the system run again,
  would the same thing happen?
• Newtonian systems Deterministic
• Relativistic systems Deterministic
• Quantum systems Indeterministic

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Are Humans Deterministic Systems?

• An argument for yes We are physical systems that
  follow deterministic laws of physics.
• Another argument for yes Suppose two patients in
  the same state are given the same drug. One dies
  and one survives. The doctor says Oh well, some
  make it and some dont. !!
• But Free Will. How could deterministic systems
  be free?

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A Probabilistic ModelPositive causal factor

• Suppose that whenever C is present, it increases
  the probability of E.
• That is, the probability of E given C is more
  than the probability of E given Not-C.
• P(EC) gt P(E-C)
• It follows that P(EC) gt P(C)
• Then C is a positive causal factor for E.

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Example

• Suppose doing exercise increases the probability
  you will live a long time.
• That is, the probability of living a long time
  given that you exercise is greater than the
  probability that you live a long time given that
  you dont exercise.
• So exercise is a positive causal factor for long
  life for you.

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Probabilistic ModelNegative Causal Factor

• Suppose whenever C is present, it decreases the
  probability of E.
• That is, the probability of E given C is less
  than the probability of E given Not-C.
• Then C is a negative causal factor for E.

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Example

• Eating fast food decreases the probability that
  you will live a long time.
• So eating fast food is a negative causal factor
  for living a long time.

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Effectiveness in Individuals

• So far weve just looked at whether C causes E or
  not.
• But some causes are more effective than others.
• The degree of effectiveness is
• P(EC) P(E-C)
• This ranges from 1 to -1.
• Maximal effectiveness is 1 0 1
• No effectiveness is P(EC) P(E-C)
• Maximal ineffectiveness is 0-1 -1

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• Suppose the probability you will live a long time
  given that you eat fast food is 0.3.
• And the probability that you will live a long
  time given that you dont eat fast food is 0.7.
• Then the causal effectiveness of fast food for
  living a long time, for you, is 0.3-0.7
• -0.4

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Philosophical Worry

• Were talking about the probability that an
  individual lives a long time.
• But which interpretation should we use?
• We cant use the actual frequency interpretation,
  because there is no population.

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• We could use counterfactual frequency
  interpretation.
• That is, we could base the probability on what
  would happen, if the individual were put in the
  same circumstances repeatedly.
• And were neck deep in metaphysical commitments
  already i.e. knowing what would happen if I had
  C.
• Or we could talk about the strength of the causal
  link between E and C (propensity). But notice
  this doesnt really explain what we mean by
  probability.

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Causal Models for Populations (Deterministic)

• When is a factor causally relevant for a
  population?
• Short answer When it is causally relevant for
  individuals in the population.
• But how should we understand the link between
  causation for individuals and causation for
  populations?
• Long answer

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• Suppose there are individuals in U for whom C is
  positively causally relevant for E, but that
  dont have C.
• Then if every individual had C, more of the
  population would have E.
• Similarly, if none of them had C, less of the
  population would have E.
• The causal connection for populations is based on
  these hypothetical populations.

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Example

• Suppose there are individuals for whom exercise
  is a positive causal factor for a long life. If
  we altered the population to one in which
  everyone exercised (C), more of the population
  would live a long life (E).
• And if we altered the population to one in which
  no-one exercised (Not-C), less of the population
  would live a long life (E).

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Actual
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X
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K
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• In general, well use hypothetical populations in
  which everyone has C (call it X) and in which
  no-one has C (call it K).
• Let PU(E) represent the proportion of the actual
  population with E.
• Let PX(E) represent the proportion of the
  hypothetical all-C population with E.
• Let PK(E) represent the proportion of the
  hypothetical all-C population without E.

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Hypothetical Population All individuals have C
X
PX(E)
Real Population
PU(E)
U
Hypothetical Population No individuals have C
PK(E)
K
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• In our example, PU(E) represents the proportion
  of the actual population that live a long life.
  Suppose this is 70.
• PX(E) represents the proportion of the
  hypothetical all-exercise population with that
  live a long life. Suppose this is 80.
• PK(E) represents the proportion of the
  hypothetical all-exercise population that live a
  long life. Suppose this is 60.

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Hypothetical Population All individuals exercise
X
PX(E) 0.8
Real Population
PU(E) 0.7
U
Hypothetical Population No individuals exercise
PK(E) 0.6
K
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• C is a positive causal factor for E in U whenever
  PX(E) is greater than PK(E).
• C is a negative causal factor for E in U whenever
  PX(E) is greater than PK(E).
• So exercise is a positive causal factor in U for
  long life.

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Negative factor

• Suppose there are individuals in U for whom C is
  negatively causally relevant for E, but that
  dont have C.
• Then if every individual had C, fewer of the
  population would have E.

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Example Negative Causal Factor for the Population

• Suppose not everyone eats fast food.
• Imagine we alter the population so that everyone
  eats fast food.
• And alter it again so no-one eats fast food.
• Then if the proportion of the all-fast-food
  population with long life were less than the
  no-fast-food population with long life, then
  eating fast food is a negative causal factor for
  long life in the population.

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Hypothetical Population All individuals eat fast
food
X
PX(E) 0.3
Real Population
PU(E) 0.7
U
Hypothetical Population No individuals eat fast
food
PK(E) 0.9
K
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Causal Relevance

• C is causally relevant for E in the population,
  U, whenever PX(E) differs from PK(E).
• C is causally irrelevant for E in the population,
  U, whenever PX(E) PK(E).

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Effectiveness in Individuals

• So far weve just looked at whether C causes E or
  not.
• But some causes are more effective than others.
• The degree of effectiveness is
• P(EC) P(E-C)
• This ranges from 1 to -1.
• Maximal effectiveness is 1 0 1
• No effectiveness is P(EC) P(E-C)
• Maximal ineffectiveness is 0-1 -1

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Effectiveness in Populations

• The measure of effectiveness in a population is
  the difference between the probability of E in
  the two hypothetical populations
• PX(E) - PK(E)
• Again, this ranges from 1 to -1.
• Maximal effectiveness is 1 0 1
• No effectiveness is P(EC) P(E-C)
• Maximal ineffectiveness is 0-1 -1

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• In our example, the effectiveness of exercise on
  long life is 0.8-0.6 0.2.
• The effectiveness of fast food on long life is
  0.3-0.9 -0.6

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Main Point

• Correlation is a relationship between properties
  in an actual population.
• Causation is a relationship between properties in
  two hypothetical populations.
• C causes E in U if the proportion of E in the
  hypothetical population with all C is greater
  than the proportion of E in the hypothetical
  population with all Not-C. 7.3

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Example of Confusion

• The Bell Curve (1994), Herrnstein Murray
• Black teachers tend to get lower scores on
  teacher competence examinations.
• Do teachers who score higher on the tests tend
  to get greater success with students?
• Study A 1 increase in teacher test scores is
  accompanied by a 5 decline in drop-out rates.
• Conclusion? Hiring teachers with higher test
  scores will reduce student drop-out rate.

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• But suppose that black teachers (who have lower
  test scores) tend to work in districts with large
  proportions of black pupils (who have higher
  failure rates).
• Then there will be a correlation between low
  scores and high failure rates.
• But there may be no causal connection between the
  two.

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• In general, teachers with good scores may go to
  the schools with the lowest failure rates.
• And teachers with low scores may go to the
  schools with the highest failure rates.
• The best students may go to the universities that
  score the highest.
• The people who choose to smoke may be disposed to
  get lung cancer.

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Exercise 7.2

• The death rate from heart attacks from widows is
  greater than among married women.
• This has been used to support the claim that
  being married prevents heart attacks.

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Widows Married
Heart attacks
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X Hypothetical population of all widows
K Hypothetical population of all married women
Is PX(E) gt PK(E)?
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Alternative non-causal hypothesis

• Widows are older.
• Being older causes heart attacks.