PHLA10F 3
Non-Deductive Arguments
PHLA10F 3
The Limits of Deduction
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Valid deductive arguments guarantee that the
conclusion is true if the premises are.
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How is this guarantee possible?
There is absolutely no more information in the
conclusion than was in the premises (usually less).
Anyone who is 2m tall is more than 1m tall.
Fred is 2m tall.
So, Fred is more than 1m tall.
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Although this argument is deductively valid, we
have thrown away some of the information
available.
All deductive arguments are like this – it is the cost
of the ‘guarantee’.
PHLA10F 3
Non-deductive arguments
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We can produce arguments that increase (or
amplify) our information only by giving up the
deductive guarantee.
Example: polling.
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Contrast
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asking everyone some question and recording their
answers
asking some of the population that question and
recording their answers
In the second case, what do we know about what the
whole population thinks?
No guarantee – but a lot less work!
PHLA10F 3
Risk and Science
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The possible gain in information from nondeductive argument is balanced by the risk of
error.
Science is risky knowledge.
Scientific theories add to our knowledge.
So there is no guarantee that they are right.
Examples:
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Newton’s theory of gravitation
Priestley’s theory of combustion.
Maxwell’s theory of electromagnetism.
Albert Einstein (1879-1955)
PHLA10F 3
Induction
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Induction is the inference from a sample of a
set of things (people’s opinions, weights,
heights, whatever) to the whole set.
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Two Crucial Factors
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Polling
Scientific testing
Sample size
Sample bias
The bigger the sample the better
The more ‘representative’ the sample the better
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representative versus random
PHLA10F 3
Induction
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A real world example
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1936 American election
the Literary Digest poll
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mailed out millions of postcard
collected 2.3 million responses
postcards were sent to
subscribers and telephone
owners
LD predicted Alf Landon would
win by a landslide
In fact, Franklin D. Roosevelt
easily won
Why was poll wrong?
Another: 2004 USA election
Franklin D. Roosevelt
Alf Landon
PHLA10F 3
Induction and Probability
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Probability measures the chance an event will
occur
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Example: an urn contains 80 red balls and 20 green
balls. After mixing them up and picking a ball out of
the urn, what is the chance it is red?
What is the chance of doing this twice and getting a
red ball both times?
Suppose we don’t know the proportion of red to
green balls in the urn.
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We can ‘poll’ the urn by picking out balls
Problem of sample size and representativeness are clear
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Suppose we only pick three balls
Suppose all the green balls are on the top and we pick there
PHLA10F 3
Abduction
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Abduction = Inference to the best explanation
Example: Gregor Mendel and the birth of
genetics
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Mendel noticed (roughly):
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green peas always produced green
offspring
yellow peas produced ¾ yellow and
¼ green in general
it was possible to make a pure yellow
‘line’ that only produced yellow
offspring
no mixed colors were produced
Johann Gregor Mendel
(1822-1884)
PHLA10F 3
Abduction
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Mendel’s Theory
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pea color depends on two factors, one from each parent
(call them G and Y)
One factor is dominant – in this case Y which means
that a pea with factors (G,Y) or (Y,G) will be yellow.
So the only way for a pea to be green is to have factors
(G,G)
There are three ways to be yellow: (Y,Y) (Y,G) (G,Y)
This theory explains Mendel’s results.
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(Y,Y) mated with (Y,Y) always gives yellow peas
(G,G) mated with (G,G) always gives green peas
‘mixed matings’ produce yellow and green in 3 to 1 ratio
PHLA10F 3
Abduction
Patterns of
Mendelian
Inheritance
PHLA10F 3
Abduction – Pathway to the Invisible
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Mendel’s theory explained his data.
This provides reason to believe his theory is
correct.
Notice that Mendel never did or even could
have seen his ‘factors’ (what we call genes).
This method is also called ‘inference to the best
explanation’
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Mendel’s theory gives a very good explanation
No other explanation does any better
Abduction is not deductive
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Despite its success Mendel’s theory could be wrong
PHLA10F 3
Abduction – The Method of Science
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Science proceeds by
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Investigating unexplained phenomena
Devising a hypothesis of theory which
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Explains the phenomena
Makes definite testable predictions
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Example: the discovery of Neptune
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Using mathematics and
Newton’s theory, Adams and Le
Verrier explained oddities in the
orbit of Uranus by
hypothesizing that there was a
more distant planet - Neptune.
Neptune-Earth Sizes
PHLA10F 3
Abduction – The Method of Science
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The Logic of Science
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Confirmation
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Is this method logically valid?
Disconfirmation
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H (hypothesis) implies O (some testable observation).
O is observed.
H is confirmed
H implies O
Not-O is observed.
H is disconfirmed or falsified.
Is this method valid?
Notice connection to the conditionals we looked at
in the deductive reasoning section
PHLA10F 3
Abduction
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The Surprise Principle
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It is too easy to make true predictions.
Hypotheses must make different predictions if they
are to be tested against each other.
Put another way: the better hypothesis makes what
we observe less surprising compared to other
hypotheses.
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Examples
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Erratic EKG.
Weightlifter.
You are flipping a coin and it comes up 25 heads in a row.
● H1: the coin is biased.
● H2: the coin is not biased (it is a ‘fair coin’).
But it would be extremely surprising if a fair coin came up heads
25 times in a row, so H1 is favoured.
PHLA10F 3
Abduction
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The Surprise Principle
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Confirmation and Probability
The surprise principle is really talking about relative
probability
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Relative Probability = the probability of an event given
another event has occurred.
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Example: what is the probability of rolling a 3 given you rolled an
odd number?
The better confirmed hypothesis is the one that
raises the probability of an observation compared to
another.
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That is, if P(E|H1) > P(E|H2) then H1 is better confirmed.
PHLA10F 3
Abduction
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What is probability?
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The measure of the chances of events
Roughly, the proportion of the target event in the set
of all possible (relevant) events
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P(rolling a 2) = 1/6 (1 target event out of 6 possibles)
P(rolling an even) = ½
Rules for combining probabilities
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P(A and B) = P(A) * P(B) (if A and B independent)
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P(A or B) = P(A) + P(B) (if A and B are exclusive)
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P(rolling 1 and then rolling 3) = 1/36
P(rolling 1 or rolling 2) = 1/3
Relative or conditional probability [P(A|B)]
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P(rolling 6 | rolled even) = 1/3
PHLA10F 3
Abduction
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The Only Game in Town Fallacy
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Abduction is not mandatory.
One can suspend judgment
When should you make an inference to the best
explanation?
When should you suspend judgment?
That depends on the plausibility of the explanatory
hypothesis and how successfully it explains the
phenomena.
Examples:
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The gremlins.
UFOs
Ghosts ....