Online notes that might be helpful

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Lecture 12
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On-line notes that might be helpful

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

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Exercise 6.13

• Is there a correlation between having a college
  education and not drinking?

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1. The Real World Population

• American adults.

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2. The Sample Data

• Among those with a college education, 75
  classified themselves as either light or moderate
  drinkers.
• 49 with a high school education gave these
  responses.

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College education High School Education
Non-drinkers or heavy drinkers
Light or moderate drinkers
0.75
0.49
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3. The Statistical Model

• The model suggested is that there is a positive
  correlation between having a college education
  and being a light or moderate drinker.

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4. Random Sampling

• How well does the study fit random sampling?
• In-home interviews.
• Random sampling
• a) All members of the population have an equal
  chance of being selected.
• Were not told how the homes are selected.
• The homeless are excluded.
• b) There is no correlation between the outcome of
  one selection and another.
• Were not told, but probably satisfied.

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5. Evaluating the Hypothesis

• Assuming random sampling, what does the data tell
  us?
• For a sample of 500 the margin of error is 4.
• Complication Were not told what proportion of
  the population had a college education.
• For n250, margin of error is-0.6

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Non-drinkers or heavy drinkers
0.81 0.69
0.55 0.42
Light or moderate drinkers
Non-drinkers or heavy drinkers
0.75
Light or moderate drinkers
0.49
n500 total
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Strength of Correlation

• 0.81-0.42 0.39
• 0.69-0.55 0.14
• So the estimated strength of correlation is
  0.39, 0.14.
• As this is based on the whole sample of 500, we
  can be 99 sure of this conclusion.

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6. Summary

• How well does the data support the evaluation of
  stage 5?
• Is the sample random enough for the hypothesis to
  be supported?
• We have to decide based on the report and the
  context in which we find the report. Its
  reasonable to assume that this was a carefully
  conducted study, in which case we have good
  evidence for the conclusion that there is a
  moderate correlation between having a college
  education and light or moderate drinking.

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Causation and Correlation

• Scientists are interested in causation, and want
  to separate causation from mere correlation.
• But is there any difference?
• And, indeed, what is causation?

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Example

• Smoking is correlated with cancer.
• Using ashtrays is correlated with cancer.
• But smoking causes cancer while using ashtrays
  does not.

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• The challenge Understanding what we mean when we
  use sentences involving cause. What makes the
  sentences true?
• Similar to the challenge for interpretations of
  probability Understanding what we mean when we
  use sentences involving probability / probably.
  What makes the sentences true?

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Common sense

• How should we understand sentences such as A
  causes B?
• Common sense A causes B means that there is
  some kind of connection between A and B
• Or
• A necessitates B
• Or
• B wouldnt have happened if A hadnt happened.
• But we dont observe this mysterious connection

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Empiricist Digression

• Logical positivists / empiricists could only
  accept phenomena they could observe.
• Connections between events are not observed
  only events are observed.
• So statements like A causes B should not be
  understood as statements about necessary
  connections.
• Nor as dependent on hypothetical situations.

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A positive suggestion

• Assume that there are no necessary causal
  connections on the world.
• The challenge now is to make sense of our use of
  the word cause.
• We use the word cause when we see a constant
  conjunction.

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Constant Conjunction

• Positive suggestion
• Hume A causes B Events of type A are
  constantly conjoined with events of type B.
• Ingesting cyanide causes death Events of
  ingesting cyanide are constantly conjoined with
  events of dying.
• Causation is constant conjunction.

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Problems with constant conjunction

• 1. Causation without constant conjunction
• 2. Constant conjunction without causation

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1. Causation without Constant Conjunction
Singular Causation

• Ducasses box. Suppose I bring in a box with a
  button on it. I press the button and the light
  goes on.
• You attribute causation to the button, even
  though you have never seen this object, or one
  like it before.
• But have you really never seen this kind of event
  before?

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2. Constant Conjunction Without Causation

• A) Two clocks. One chimes immediately before the
  other chimes.
• B) Day always follows night.

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Responses

• Perhaps the two events must be spatio-temporally
  connected. Or some process connecting them.
• But isnt action-at-a-distance conceptually
  impossible?
• Cause by absence.

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Counterfactual Theories

• Where A and B are actual events,
• A causes B If A were not to occur, then B
  would not occur.
• In the closest world in which not A, not B

_at_
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• Think about it as making the minimal changes
  necessary to make Not-A true.
• Example Make the minimal changes necessary for
  it to be true that the cyanide was not ingested.
  In that world, was there death? If not, then the
  cyanide caused the death.

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A worlds
_at_
A and B at actual world
Nearest world where not A If Not-B, then A causes
B.
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Problems with counterfactual theories 1

• Context-sensitivity.
• A camper lights a fire that gets out of control
  and burns down the forest.
• If the camper hadnt been born, the forest
  wouldnt have burned down.
• But would we want to say that the campers being
  born caused the fire?

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Problems with counterfactual theories 2

• Transitivity.
• According to counterfactual theories, causation
  is transitive If A causes B, and B causes C,
  then A causes C.
• But causation is not transitive.

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• Suppose a subject gets shocked when Al and Bobs
  switches are in the same position.
• They start off in the right position.
• Al doesnt want to shock the subject, so he moves
  his switch to the left.
• Bob dislikes the subject, so moves his switch to
  the left.
• Als move caused Bobs move, which caused the
  shock.
• But do we want to say that Al caused the shock?

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Problems with counterfactual theories 3

• Suzy and Bill both throw rocks at a pane of
  glass.
• Suzys hits first, and shatters the glass.
• But if Suzys rock had not shattered the glass,
  Bills would have.
• So according to the counterfactual theory it is
  not true that Suzys throwing the rock causes the
  glass to break.