Inductive Reasoning
The Nature of Inductive
Reasoning

What is an inductive argument?
1.
Any argument which is not deductive!
I.e., any argument which does not
provide a guarantee of the truth of the
conclusion if the premises are true.
Inductive arguments are probabalistic.


The Nature of Inductive
Reasoning

What else?
Inductive arguments are ampliative, while
deductive arguments are non-ampliative.
 An argument is ampliative (defn) iff there is
information contained in the conclusion that is
not already contained in the premises.
 That’s the trade-off! You lose the guarantee of
the truth of the conclusion for amplification.
2.
The Nature of Inductive
Reasoning
3.
The logical strength of inductive
arguments is not dependent on the form
of the argument, but rather the content
of the premises. (It’s the opposite for
deductive arguments.)
The Nature of Inductive
Reasoning

1.
2.
3.
4.
However, there are 4 forms of inductive
arguments that usually regarded as
logically strong so long as certain
conditions are met.
Inductive generalization
Statistical syllogism
Induction by confirmation
Analogical reasoning
Inductive Generalization

Has the following form:
Z percent of observed F’s are G
It is probable, therefore, that Z percent of
all F’s are G.
Inductive Generalization

E.g.

60% of students at STFX who were
questioned believe in God. It is probable,
therefore, that 60% of students at STFX
believe in God.
Inductive Generalization

When assessing these arguments, ask:
1.
Is the sample representative?
Is the sample large enough?
2.
Statistical Syllogism

Has the following form:
Z percent of all F’s are G
x is an F
Is it probable to the degree 0.Z that x is G
Statistical Syllogism

What’s the difference between an inductive
generalization and a statistical syllogism?
Inductive generalizations reason from
particular observations to a general claim
about a class.
 Statistical syllogisms reason from a general
claim about a class to a claim about a
particular individual.

Statistical Syllogism

E.g.,
60% of students at STFX believe in God.
Bob is a student at STFX.
Therefore, there is a .6 degree of probability
that Bob believes in God.
Statistical Syllogism

When assessing these arguments, ask:
1.
Is there any additional information about x that
has not been included in the premises?
E.g. Bob is President of Catholic League of
Students (prob that he believes in God
increases).
 E.g., Bob is President of the Atheists for the
Environment Society (Prob that he believes in
God decreases).

Induction by Confirmation
Induction can be used to support a hypothesis
or theory by providing confirming instances of
that hypothesis or theory.
 When we propose a theory or hypothesis, there
are certain things that ought to be observed if it
is actually true (or probable).
 These are called observation statements. If we
do observe what the theory predicts, then we
have confirmed the theory.

Induction by Confirmation

Induction by Confirmation then has the following
form:
If h then o
o
It is probable that h
NB: similar to the formal fallacy of “affirming the
consequent”
Induction by Confirmation

E.g.
If the theory of general relativity is true, then it
follows that light rays passing near the sun will
bend.
During the solar eclipse of 1919 it was observed
that light rays passing near the sun did bend.
It is probable therefore that the theory of general
relativity is true.
Induction by Confirmation

When assessing these arguments, ask:
1.
Is the number of confirming instances
relatively high?
•
2.
In general, the more confirming instances
the better the theory.
Are there any disconfirming instances?
•
Any disconfirming instance refutes the
theory.
Induction by Confirmation

Disconfirming instances are regarded as refutations of a
theory because such a refutation takes this form:
If h then o
Not-o
Therefore not-h

That is, a disconfirming instances refutes a theory
because we are dealing with a deductively valid
argument form: Modus Tollens (denying the
consequent).
Analogical Reasoning

Analogical reasoning works by comparing
things which are similar (analogous) and
concluding that properties or relations that
one thing has must also be present in the
other.
Analogical Reasoning

E.g.,
Last year I put some fertilizer on my
strawberries and in the fall got about 20
per cent more strawberries. You should
do the same with your strawberries, since
you got the same kind of soil. You’ll
probably get more strawberries too.
Analogical Reasoning
Analogies compare two cases: the
subject case, and the analogue case.
 The subject case is the case about which
we are trying to derive a conclusion
(fertilizer on your soil)
 The analogue case is the case about
which we are more familiar (fertilizer on
my soil).

Analogical Reasoning
The conclusion in an analogy makes a
claim about the subject case, and in
particular states that the subject case will
(probably) have the target feature.
 The target feature (increase in strawberry
production) is the feature that is present
in the analogue case, and it is being
concluded that it (probably) is in the
subject case.

Analogical Reasoning

1.
2.
There are two kinds of analogical
arguments:
Analogical Argument by Properties
Analogical Argument by Relations
Analogical Argument by
Properties

Analogical Argument by Properties has the
following form:
x has A, B, C. [analogue case]
y has A, B.
[subject case]
It is probable therefore that y has C [target
feature]
Analogical Argument by
Properties
E.g.,
Canada geese are water birds that nest in
Canada in the early spring and migrate
south to warmer climates for the winter
months. Ducks are also water birds that
nest in Canada in early spring. Therefore,
ducks probably migrate south for the
winter, too.

Analogical Argument by
Properties
P1 [analogue case]: Canada geese are water birds
that nest in Canada in the early spring and
migrate south to warmer climates for the winter
months.
P2 [subject case]: Ducks are also water birds that
nest in Canada in early spring.
Conclusion: Therefore, ducks probably migrate
south for the winter [target feature], too.
Analogical Argument by
Relations

Analogical Argument by Relations has the
following form:
x is to y [analogue case] as a is to b
[subject case].
x is R to y.
It is probable therefore that a is R to b
[target feature]
Analogical Argument by
Relations
E.g.,
The proposal to give clean needles to prison
inmates to stop the spread of AIDS from
the use of dirty needles is ridiculous. It is
like giving bank robbers normal bullets to
stop them from using dum-dum bullets,
which are much more damaging to the
victim.

Analogical Argument by
Relations
P1: Dum-dum bullets are to normal bullets (as used by
bank robbers) [analogue case] as dirty needles are to
clean (as used by prison inmates) [subject case].
P2: Although dum-dum bullets are much more damaging to
the victim, normal bullets still kill their victims. Further,
the role of police officers is to stop bank robbers, not
prevent the harms they cause.
Conclusion: Although dirty needles are more damaging to
the victim (addicts are likely to get HIV, etc.), clean
needles can be just as damaging (e.g., overdoses).
Further, the role of prison officials is to stop drug use,
not prevent the harms caused by it.
Analogical Reasoning

When assessing these arguments, ask:
1.
2.
3.
4.
How many entities are we comparing?
What is the variety of dissimilarity?
In how many respects are the entities similar?
Are the respects in which the compared
entities are similar relevant to the conclusion?
In what ways are the entities under
consideration dissimilar?
How bold or modest is the conclusion?
5.
6.