Inductive Logic

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 donít 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 yearís 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 doesnít 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 dayís date from one year to another.Or is there a connection?Letís 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 canít 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.Letís 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 didnít 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 donít then we would be forced to modify our conclusion.We would have a case where itís 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.