Arguments
in Inductive Logic
Ken Wais 4/16/14
Methodologies of
Inductive and Deductive Logic
Inductive
logic, unlike deductive logic develops its conclusions by examining particular
facts and attempts to extend these facts to a conclusion about all cases for
which the facts are applied. Deductive
logic is the direct opposite of the previous notion. Deductive logic is not concerned with truth
or falsity of facts, but whether the method used to derive the conclusion is
valid. In this sense, one can derive a
valid deductive argument that is factually false. We could show a syllogism about imaginary
creatures or objects. Though, these
items don’t exist, the conclusions about them would be valid. Induction logic is not applied in this
way. Inductive logic seeks to apply
deductive methods to material facts. So,
an inductive argument might use syllogism to determine if a series of
observations leads to a certain conclusion.
This form of logic has been called reasoning from the particular to the general. Given some sample
of facts, conclusions are extended to the general case concerning those facts. This is the principle difference between
deductive and inductive methodology.
Induction in science attempts to apply the deductive method of
reasoning, by using information derived from part of a whole collection of
elements. Statistical analysis is a
field in science that does just that.
About the Argument
Below
we will examine an argument that is advanced based on an empirical
finding. The argument will be evaluated
as valid or fallacious based whether its conclusion is true or false. We will state a fact that has been
established from statistical tabulation events over a period of 10 years. The basis for this argument will be
inductive. That is, we will reason from
the part to the whole.
The Fact
Based
on statistical tabulation over a period of 10 years, actuarial statisticians
have found that the probability of bodily injury accidents occurring in the
home is 75% greater on Wednesday, than any other day of the week.
The Hypothesis
Can
the fact cited above be extended to show that for every DATE that is a
Wednesday with 75% more accidents occurring in a given year, the probability of
it occurring in the following year is 75%?
This is a more complex problem than primer facie it appears. An example will illustrate this. In the year 2000 the Wednesday of May 10,
2000 occurred. In the year 2001 the
Wednesday of January 10, 2001 occurred.
The date is the same not the month.
So, will there be the same number of dates for Wednesday 2001 as
2000? The only way to know is to enumerate
them for the 2 years.
Analytic Examination of
the Problem
Clearly
this is a hypothesis fit for inductive logic.
We merely have to determine if it is true or false. One method of inductive logic not mentioned
above is enumeration. Enumeration would
be to count every date that is a Wednesday in one year (which we know have 75%
probability of being accident days) and compare those dates to the next and see
if the next year’s date have the 75% probability. Enumeration is exhaustive and is usually not
used to arrive at an induction conclusion.
But, in this case we can do just that since the specification doesn’t
involve extreme large numbers of comparisons.
In this case we are considering 120 months or 10 years of data. We can count the number of Wednesdays in this
collection of months and see if the DATE of the Wednesdays is 75% of the
accidents that occur year-to-year. This
would be going from the part (the days of the week) to the whole (the dates of
the year). Right away, we can see this
may be a fallacious conclusion since there is not a material connection between
a day’s date from one year to another. Or is there a connection? Let’s find out.
Conclusion
The
answer is no. After looking at the dates
of Wednesdays from 2000 to 2001 and tabulating their occurrences, the dates are
not the same. So, the 75% probability
can’t apply to dates as it does to Wednesdays from year-to-year. Notice we only looked at one year-to-year
period. We examined a part of the whole
in other words. Let’s restate this using
deductive symbolism.
D
= Probability of bodily injury accidents in the home is 75% greater on
Wednesdays of any given year.
d=
Probability of bodily injury accidents in the home is 75% greater on the dates
that are Wednesday of any given year.
The
implication is D→d. We can restate
this using the values for D and d, as shown below:
If
(Probability of bodily injury accidents in the home is 75% greater on
Wednesdays of any given year), Then (Probability of bodily injury accidents in
the home is 75% greater on the dates that are Wednesday of any given year).
Using
Modus Ponens (MP) we have the
following truth table for the above implication:
D |
d |
D→d |
T |
T |
T |
T |
F |
F |
F |
T |
T |
F |
F |
T |
Third
row is the case we found inductively in our enumeration. That is, the first premise is true while the
second premise is false and thus the whole implication is false. This follows from the form of MP
reasoning. Stated literally it means we
can never argue from true premises to false conclusions. This is exactly what we found when we counted
the number of dates that are Wednesdays from the year 2000 to the year
2001. The dates didn’t match the
previous year, so corresponding probabilities for the dates would not match the
day of the week. This may seem to be an obvious
fallacious conclusion. Yet, it is not
that clear-cut. We have only sampled 10
years of data. Suppose we sampled 100
years of data. It is possible that we
would find that in that data set many years repeat the exact same number of
dates as previous years. In that case,
would have a valid MP argument. Why? The
two years sampled would have the exact same date and day of the week
samples. The MP row would the first in
this case and the argument would be valid.
If we looked at the 100 year sample set and found that some years the
dates matched and other they don’t then we would be forced to modify our conclusion. We would have a case where it’s inconclusive if the MP argument is valid
without more source data
This
brings us to an important consideration about inductive logical conclusions in
science:
Inductive
logical derivations are only as good as the empirical data used to arrive at
their conclusions.
That
is inductive conclusions can be modified and even reversed if more
comprehensive data are collected and analyzed.
This in turn, means that inductive logical reasoning never gives us
absolute knowledge of the subject studied.
This is not a disappointing fact.
It signifies that inductive methods can be a starting point for making
actionable decisions. In the case above
if the 100 data were further refined it might be the case that certain cycles
appear in which sometimes the date/date of the week data are identical year-to-year
and in others not. The conclusions could
be modified to apply to only certain cycles of years. Insurance companies would make good use of
this knowledge when deciding on bodily injury compensation limits among other
things.