Induction
CIS308
Dr Harry Erwin
Inductive reasoning
• Inductive reasoning is the complement of
deductive reasoning.
• Induction or inductive reasoning, sometimes
called inductive logic, is the process of reasoning
in which the premises of an argument are believed
to support the conclusion but do not ensure it.
• It is used to ascribe properties or relations to types
based on tokens (i.e., on one or a small number of
observations or experiences); or to formulate laws
based on limited observations of recurring
phenomenal patterns.
Propositions
• Induction is used, for example, in using
specific propositions such as:
– This ice is cold.
– A billiard ball moves when struck with a cue.
• ...to infer general propositions such as:
– All ice is cold.
– All billiard balls struck with a cue move.
Hume
• Inductive reasoning has been attacked several
times.
• Historically, David Hume denied its logical
admissibility.
• During the 20th century, most notably Karl Popper
and David Miller have disputed the existence,
necessity and validity of any inductive reasoning,
even of probabilistic (bayesian) ones.
Strong induction
– All observed crows are black.
– therefore,
– All crows are black.
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This exemplifies the nature of induction: inducing the universal from the
particular. However, the conclusion is not certain.
Unless we can systematically falsify the possibility of crows of another color,
the statement (conclusion) may actually be false.
For example, one could examine the bird's genome and learn whether it's
capable of producing a differently colored bird without mutation or a long set
of breeding changes. In doing so, we could discover that albinism is possible,
resulting in light-colored crows.
Even if you change the definition of "crow" to require blackness, the original
question of the color possibilities for a bird of that species would stand, only
semantically hidden.
A strong induction is thus an argument in which the truth of the premises
would make the truth of the conclusion probable, but not definite.
Weak induction
– I always hang pictures on nails.
– therefore
– All pictures hang from nails.
• Assuming the first statement to be true, this example is built on the
certainty that "I always hang pictures on nails" leading to the generalization
that "All pictures hang from nails". However, the link between the premise
and the inductive conclusion is weak. No reason exists to believe that just
because one person hangs pictures on nails that there are no other ways for
pictures to be hung, or that other people cannot do other things with
pictures. Indeed, not all pictures are hung from nails; moreover, not all
pictures are hung. The conclusion cannot be strongly inductively made
from the premise. Using other knowledge we can easily see that this
example of induction would lead us to a clearly false conclusion.
Conclusions drawn in this manner are usually overgeneralizations.
Weak induction
– Many speeding tickets are given to teenagers.
– therefore
– All teenagers drive fast
• In this example, the premise is built upon a certainty;
however, it is not one that leads to the conclusion. Not
every teenager observed has been given a speeding ticket.
In other words, unlike "The sun rises every morning", there
are already plenty of examples of teenagers not being
given speeding tickets.
• Therefore the conclusion drawn can easily be true or false
(perhaps more easily false than true in this case), and the
inductive logic does not give us a strong conclusion.
Weak induction
• In both of these examples of weak
induction, the logical means of connecting
the premise and conclusion (with the word
"therefore") are faulty, and do not give us a
strong inductively reasoned statement.
Validity
• Formal logic, as most people learn it, is deductive rather than
inductive. Some philosophers claim to have created systems of
inductive logic, but it is controversial whether a logic of induction is
even possible.
• In contrast to deductive reasoning, conclusions arrived at by inductive
reasoning do not necessarily have the same degree of certainty as the
initial premises.
• For example, a conclusion that all swans are white is false, but may
have been thought true in Europe until the settlement of Australia,
when Black Swans were discovered.
• Inductive arguments are never binding but they may be cogent.
• Inductive reasoning is deductively invalid.
• (An argument in formal logic is valid if and only if it is not possible for
the premises of the argument to be true whilst the conclusion is false.)
Validity
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In induction there are always many conclusions that can reasonably be related to
certain premises.
Inductions are open; deductions are closed.
It is however possible to derive a true statement using inductive reasoning if you
know the conclusion.
The only way to have an efficient argument by induction is for the known
conclusion to be able to be true only if an unstated external conclusion is true, from
which the initial conclusion was built and has certain criteria to be met in order to
be true (separate from the stated conclusion).
By substitution of one conclusion for the other, you can inductively find out what
evidence you need in order for your induction to be true.
For example, you have a window that opens only one way, but not the other.
Assuming that you know that the only way for that to happen is that the hinges are
faulty, inductively you can postulate that the only way for that window to be fixed
would be to apply oil (whatever will fix the unstated conclusion).
From there on you can successfully build your case.
However, if your unstated conclusion is false, which can only be proven by
deductive reasoning, then your whole argument by induction collapses.
Thus ultimately, pure inductive reasoning does not exist.
Validity
• Hume highlighted the fact that our everyday reasoning
depends on patterns of repeated experience rather than
deductively valid arguments.
• For example, we believe that bread will nourish us because
it has done so in the past, but this is not a guarantee that it
will always do so.
• As Hume said, someone who insisted on sound deductive
justifications for everything would starve to death.
• Instead of approaching everything with unproductive
skepticism, Hume advocated a practical skepticism based
on common sense, where the inevitability of induction is
accepted.
Validity
• Induction is sometimes framed as reasoning about the future from the
past, but in its broadest sense it involves reaching conclusions about
unobserved things on the basis of what has been observed.
• Inferences about the past from present evidence – for instance, as in
archaeology, count as induction.
• Induction could also be across space rather than time, for instance as in
physical cosmology where conclusions about the whole universe are
drawn from the limited perspective we are able to observe.
• Twentieth-century philosophy has approached induction very
differently. Rather than a choice about what predictions to make about
the future, induction can be seen as a choice of what concepts to fit to
observations or of how to graph or represent a set of observed data.
Nelson Goodman posed a "new riddle of induction" by inventing the
property "grue" to which induction does not apply.
Types of inductive reasoning
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Generalization
Statistical syllogism
Simple induction
Argument from analogy
Causal inference
Prediction
Argument from authority
Bayesian inference
Generalization
• A generalization (more accurately, an inductive
generalization) proceeds from a premise about a sample to
a conclusion about the population.
– The proportion Q of the sample has attribute A.
– therefore
– The proportion Q of the population has attribute A.
• How great the support which the premises provide for the
conclusion is dependent on (a) the number of individuals
in the sample group compared to the number in the
population; and (b) the randomness of the sample.
• The hasty generalization and biased sample are fallacies
related to generalization.
Statistical syllogism
• A statistical syllogism proceeds from a generalization to a
conclusion about an individual.
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A proportion Q of population P has attribute A.
An individual I is a member of P.
therefore
There is a probability which corresponds to Q that I has A.
• The proportion in the first premise would be something
like "3/5ths of", "all", "few", etc.
• Two dicto simpliciter fallacies can occur in statistical
syllogisms: "accident" and "converse accident".
Syllogism
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A syllogism consists of three parts:
the major premise, the minor premise, and the conclusion.
In Aristotle, each of the premises is in the form:
"Some/all A belong to B," where "Some/All A' is one term and
"belong to B" is another, but more modern logicians allow some
variation. Each of the premises has one term in common with the
conclusion: in a major premise, this is the major term (i.e., the
predicate) of the conclusion; in a minor premise, it is the minor term
(the subject) of the conclusion.
• For example:
– Major premise: All humans are mortal.
– Minor premise: Socrates is human.
– Conclusion: Socrates is mortal.
Simple induction
• Simple induction proceeds from a premise about a sample
group to a conclusion about another individual.
– Proportion Q of the known instances of population P has attribute
A.
– Individual I is another member of P.
– therefore
– There is a probability corresponding to Q that I has A.
• This is a combination of a generalization and a statistical
syllogism, where the conclusion of the generalization is
also the first premise of the statistical syllogism.
Argument from analogy
• An (inductive) analogy proceeds from known similarities
between two things to a conclusion about an additional
attribute common to both things.
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P is similar to Q.
P has attribute A.
therefore
Q has attribute A.
• An analogy relies on the inference that the properties
known to be shared (the similarities) imply that A is also a
shared property. The support which the premises provide
for the conclusion is dependent upon the relevance and
number of the similarities between P and Q. The fallacy
related to this process is false analogy.
Causal inference
• A causal inference draws a conclusion
about a causal connection based on the
conditions of the occurrence of an effect.
Premises about the correlation of two things
can indicate a causal relationship between
them, but additional factors must be
confirmed to establish the exact form of the
causal relationship.
Prediction
• A prediction draws a conclusion about a
future individual from a past sample.
– Proportion Q of observed members of group G
have had attribute A.
– therefore
– There is a probability corresponding to Q that
other members of group
– G will have attribute A when next observed.
Argument from authority
• An argument from authority draws a conclusion about the
truth of a statement based on the proportion of true
propositions provided by an authoritative source. It has the
same form as a prediction.
– Proportion Q of the claims of authority A have been true.
– therefore
– There is a probability corresponding to Q that this claim of A is
true.
• For instance:
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All observed claims from websites about logic are true.
Information X came from a website about logic.
therefore
Information X is likely to be true.
Bayesian inference
• Of the candidate systems for an inductive logic,
the most influential is Bayesianism. This uses
probability theory as the framework for induction.
• Given new evidence, Bayes' theorem is used to
evaluate how much the strength of a belief in a
hypothesis should change.
• There is debate around what informs the original
degree of belief. Objective Bayesians seek an
objective value for the degree of probability of a
hypothesis being correct and so do not avoid the
philosophical criticisms of objectivism.
Bayesian inference
• Subjective Bayesians hold that prior probabilities represent
subjective degrees of belief, but that the repeated
application of Bayes' theorem leads to a high degree of
agreement on the posterior probability.
• They therefore fail to provide an objective standard for
choosing between conflicting hypotheses. The theorem can
be used to produce a rational justification for a belief in
some hypothesis, but at the expense of rejecting
objectivism. Such a scheme cannot be used, for instance, to
decide objectively between conflicting scientific
paradigms.
Subjectivity?
• Edwin Jaynes, an outspoken physicist and Bayesian, argued that
"subjective" elements are present in all inference, for instance in
choosing axioms for deductive inference; in choosing initial degrees of
belief or prior probabilities; or in choosing likelihoods.
• He thus sought principles for assigning probabilities from qualitative
knowledge.
• Maximum entropy – a generalization of the principle of indifference –
and transformation groups are the two tools he produced.
• Both attempt to alleviate the subjectivity of probability assignment in
specific situations by converting knowledge of features such as a
situation's symmetry into unambiguous choices for probability
distributions.
Inductive logic?
• Cox's theorem, which derives probability
from a set of logical constraints on a system
of inductive reasoning, prompts Bayesians
to call their system an inductive logic.