SPEAKER 1: In the terminology
of standard textbook logic
an inductive argument
is one that's intended
to be strong rather than valid.
But this terminology
isn't necessarily
standard outside of logic.
In the sciences in
particular there
is a more common
reading of induction
that means something
like making an inference
from particular cases
to the general case.
In this tutorial,
we're going to talk
about this reading
of the term and how
it relates to the
standard usage in logic,
and the role of inductive
reasoning in the sciences
more broadly.
In the sciences,
the term induction
is commonly used to
describe inferences
from particular cases
to the general case,
or from a finite sample of
data to a generalization
about a whole population.
Here's the prototype
argument form.
It illustrates this notion
of inductive inference.
You note that some individual
of a certain kind, a1,
has a property, B. An example
would be this swan is white.
Then you note that some other
individual of the same kind,
a2, has the same property.
This other swan is white.
And you keep doing this
for all the individuals
available to you.
This swan is white.
And this swan is white.
And this swan over there
is white, and so on.
So you've observed n swans
and all of them are white.
From here the inductive
move is to say
that all swans
everywhere are white,
even the swans that you
haven't observed and will never
observe.
This is an example of an
inductive generalization.
And arguments of
this form or that
do something similar, namely
infer a general conclusion
from a finite example, exemplify
the way the term induction is
most commonly used in science.
Another example that
illustrates inductive reasoning
in this sense is the
reasoning involved
in inferring a functional
relationship between two
variables based on a
finite set of data points.
Let's say you heat a metal rod.
You observe that it
expands the hotter it gets.
So for various
temperatures you plot
the length of the rod
against the temperature.
You get a spread of data
points that looks like this.
What you may want
to know, though,
is how length varies with
temperature generally
so that for any
value of temperature
you can then predict
the length of the rod.
To do that you might try
to draw a curve of best
fit through the data
points, like so.
Looks like a pretty
linear relationship,
so a straight line
with a slope like this
seems like it'll give
a good fit to the data.
Now, given the equation for
this functional relationship,
you can now plug in a
value for the temperature
and derive a value
for the length.
The equation for
the straight line
is an inductive
generalization that you've
inferred from the finite
set of data points.
The data points, in fact,
are functioning as premises,
and the straight line is
the general conclusion
that you're inferring
from those premises.
When you plug in a value
for the temperature
and derive a value for
the length of the rod
based on the equation
for the straight line,
you're driving a prediction
about a specific event
based on the generalization
you've inferred from the data.
This example illustrates
just how common inductive
generalizations are in science.
So it's not surprising
that scientists have a word
for this kind of reasoning.
In fact, the language of
induction used in this sense
can be traced back to people
like Francis Bacon, who
back in the 17th century
articulated and defended
this kind of reasoning as
a general method for doing
science.
So how does this
kind of reasoning
relate to the definition
of induction used in logic?
We'll recall in standard
logic, an argument
is inductive if it's intended
to be strong rather than valid.
The key thing to
note is that this
is a much broader definition
than the one commonly used
in the sciences.
That definition focuses on
arguments of a specific form--
those where the premises are
claims about particular things
or cases and the conclusion
is a generalization
from those cases.
But if you take the
standard logical definition
of an inductive
argument, you find
that many different
kinds of arguments
will qualify as inductive
not just arguments
that infer generalizations
from particular cases.
So for example, on
the logical definition
a prediction about the
future based on past data
will count as an
inductive argument.
The sun has risen every day
for as long as the earth has
existed, as for as we know.
So we expect the sun to
rise tomorrow as well.
This is an inductive
argument on our definition
because we acknowledge that
even with this reliable track
record, it's still possible for
the sun to not rise tomorrow.
Aliens, for example,
might blow up
the earth or the sun overnight.
So this inference from the past
to the future is inductive,
and most of us would say
that it's a strong inference
But notice that it's not an
argument from the particular
to the general.
The conclusion isn't
a generalization.
It's a claim about
a particular event--
the rising of the sun tomorrow.
So this kind of argument
wouldn't count as inductive
under the standard
science definition,
but it does count under the
standard logical definition.
Here's a second example that
illustrates the difference.
90% of human beings
are right-handed.
John is a human being,
therefore John is right-handed.
Notice that the main
premise is a general
claim while the
conclusion is a claim
about a particular person.
On the standard
science definition
this isn't an inductive
argument since it's
moving from the general
to the particular
rather than from the
particular to the general.
But on the logical
definition of induction
this argument does count
since the argument is intended
to be strong not valid.
The relationship between the
two definitions looks like this.
The arguments that
qualify as inductive
under the standard
science definition
are a subset of
the arguments that
qualify as inductive under the
standard logical definition.
So from a logical
point of view, there's
no problem with
calling an inference
from the particular to the
general an inductive argument
since all such arguments satisfy
the basic logical definition.
But scientists are
sometimes confused
when they see the
term induction used
to describe other forms
of reasoning than the ones
that they normally associate
with inductive inferences.
There shouldn't be
any confusion as long
as you keep the
two senses in mind
and distinguish them
when it's appropriate.
But if you don't
distinguish them,
then you may run into
discussions like this one that
contradict themselves.
These are the
first few sentences
of the Wikipedia
entry on induction
at the time of
making this video.
The first sentence presents the
standard logical definition--
inductive reasoning is defined
as strong reasoning, reasoning
that doesn't guarantee truth.
The second sentence presents
the standard science definition
of induction, defining
it as reasoning
from the particular
to the general.
Later on in the
article, they present
a number of examples
of inductive arguments
that satisfy the
logical definition
but not the scientific
definition, such as inferences
from correlations to causes, or
predictions about future events
based on past events, and so on.
These examples they use flat
out contradict the definition
highlighted in yellow here.
So if anyone out
there's inclined,
you might want to
edit that entry
to clarify the distinction
we've been discussing here.
I'll summarize some key
points of this discussion.
The first is that
we should be aware
that there's a difference
between the way the term
induction is defined in general
scientific usage and the way
it's defined in logic.
The logical definition
is much broader
and is basically synonymous
with non-deductive inference.
The scientific usage
is narrower and it
focuses on inferences from
the particular to the general.
Second, induction in the
broader logical sense
is fundamental to scientific
reasoning in general.
That Inductive reasoning
is risky reasoning.
It's fallible reasoning.
Are you moving from known facts
about observable phenomena,
say, to a hypothesis or
conclusion about the world
beyond the observable facts?
And the distinctive feature
about this kind of reasoning
is that you could have all
the observable facts right,
but you can still be wrong about
the generalizations you draw
from those observations
or the theoretical story
you tell that tries to
explain those observations.
It's a fundamental feature
of scientific theorizing
that it's revisable in light
of new evidence and experience.
It follows from this observation
that scientific reasoning
is, broadly speaking,
inductive reasoning,
that scientific
arguments should aim
to be strong rather
than valid, that it's
both unrealistic and confused
to expect them to be valid.
The disciplines that
trade in valid arguments
and valid inferences are
fields like mathematics,
computer science, and
formal deductive logic.
The natural and social
sciences, on the other hand,
deal with fallible,
risky inferences
and they aim for
strong arguments.
