INDUCTIVE AND DEDUCTIVE REASONING

Simple logic was a favorite topic in my Philosophy degree, so my enthusiasm for the Geometry chapters that deal with reasoning can be easily explained.  The roots that are developed through the Geometry curriculum can be used in many other contexts: debate, essay writing, solving problems, and even convincing your parents to extending curfew.  In Geometry class right about now, if logic precedes introduction of two-column proofs, we should be thinking about inductive and deductive reasoning.

INDUCTIVE REASONING goes from the specific to the general, from observed details to global conjectures.  It’s what a scientist would use to develop a hypothesis after collecting data, discovering a pattern, and assuming that the pattern will continue.  It’s also the thinking that is expected on the last couple of questions in any data set on the ACT Science Reasoning section.

A determining factor of inductive reasoning is that conjectures based on an infinite universe can never be proven.  We can only DISPROVE them.  Let’s say a statistician collects a huge amount of data about  monkeys in a maze.  The data shows that every time a monkey consumed carrots before running the maze, the elapsed time was shorter than without eating carrots.  This pattern continued with tens of thousands of monkeys.  The scientist hypothesizes that eating carrots will always improve a monkey’s speed in running a maze.  It would be impossible to test every monkey, so we could never prove the hypothesis is universally true.  But find just one monkey whose performance did NOT improve and the hypothesis is disproven.

The same principle of finding a counterexample will be helpful later when we get to indirect proof.

DEDUCTIVE REASONING (general to specific) starts with a global statement, properties, definitions, postulates, or theorems and applies them to form conclusions in specific situations.  It assumes that what is true of the large group is true of all of the members of the group.  This is the form of reasoning we’ll be using in two-column proof in Geometry.

Within deductive reasoning, we find two important principles: the laws of DETACHMENT and SYLLOGISM.

                 DETACHMENT follows my proposal that mathematicians aren’t poets engaging in metaphors.  When mathematicians mean detach, they say detach.  So the Law of Detachment says that something can be removed or detached.

If P implies Q
and P is true (given)
then Q is also true.

Q can be detached from the original statement and concluded to be true. This is a primary construction in Geometry  proofs.

1.  If a polygon has three sides, then it is a triangle.
2.  This polygon has three sides.
3.  Therefore, it is a triangle.

                  SYLLOGISM:  If you learned the transitive property in Algebra, you already know about syllogisms.

The placement of P, Q, and R are important in syllogisms.  R is the conclusion you want to come to and is considered the MAJOR term.  P is called the MINOR term, and Q is the MIDDLE term.  As “transitive” implies, we’re moving over or across the Q to get P to relate to R.



VALIDITY AND SOUNDNESS

CAUTION!  When we’re talking about logical steps to conclusions, we need to distinguish between validity and soundness.  If an argument is VALID, it follows the syllogistic steps already listed.  Validity is determined by the form of the argument and says nothing about truth, either of the premises or the conclusion.  A conclusion can be valid but unsound.

If the premises are not true, even if they follow appropriate steps, then the conclusion is not sound.  In order for an argument to be SOUND, the proper logical steps must be followed and premises must also be true. 

Here’s an example of a valid argument which is NOT sound.

    All my pets are mammals.
    Lizzie the Lizard is my pet.
    Therefore, Lizzie the Lizard is a mammal.  (Valid, but false: a lizard is not a mammal. The conclusion is not sound because the first premise is not true.)

Some students try to determine the validity of a conclusion by deciding if it and the premises are true.  This is a slippery slope in logic.  A better approach is to put premises into the “P implies Q” format and look at the steps used to arrive at the conclusion.

This example is not even valid because the steps are wrong, but the conclusion could be either true or false:

P, Q, and R statements might go like this:
    All dogs are mammals.
    All cats are mammals.
    All dogs are cats.  (Not Valid because the steps are wrong, and not Sound even though both premises are true, and obviously false.)

Looking only at the truth of the conclusion to determine its soundness can lead to mistakes.
    All mammals are born alive.
    All elephants are born alive.
    Therefore, all elephants are mammals.  (The conclusion is true, but the logic is not valid because it does not follow the Law of Syllogism, and the conclusion is not sound because the first premise is not true.  The platypus is not a mammal but offspring are born alive.)

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SUMMARY FOR YOUR LOGIC STUDY CARD:

Inductive Reasoning: specific to general

Deductive Reasoning: general to specific
     Detachment: If A, then B.  A.  Therefore, B.
     Syllogism:  If A, then B.  If B, then C.  Therefore, If A, then C.

Validity:  Are the Laws of Detachment and Syllogism followed?

Soundness:  Are the premises valid AND true?

TRUTH TABLES

In arithmetic and algebra, we are  working largely with numbers.  When Geometry comes into play, a lot of the work involves words to convey theorems and postulates.  “If a triangle has three congruent angles, then it is an equiangular triangle.”  Try writing THAT with X and Y!!  So now we need a way to work with these statements and to know when a statement is true.

You’ve already learned some things about an “if-then” conditional statement. Remember converse, inverse, and contrapositive?  For truth tables, we’ll call the first part of the conditional statement "P" and the second part "Q."  I can’t think of a specific reason for selecting these letters, but they are the ones other mathematicians use, so why not? 

For a conditional statement like “If two angles form a straight line, then the angles are supplementary,”  P will be the “two angles form a straight line” part and Q will be “the angles are supplementary” part.  Notice that "if" and "then" are detached.  They always stay in the same order, but we'll move the P and Q parts around.

The original statement can be summarized as “If P, then Q.”   For further brevity, we can write

and read it as “P implies Q.”

If P is true, then not-P ( -P) is false.  And if Q is true, then not-Q (-Q) is false.
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Now for the TRUTH TABLES.  There are four ways to judge the conditional statement.  The first part (P) can be true and the second part (Q) either true or false.  Or the first part can be false while the second part is either true or false.  The truth table looks like this:

Notice that the statement is only false under one scenario:  when the first part is true but the second part is not.  That may seem counterintuitive, so let’s look at it closely.


Let’s think of a more personal conditional statement “If you miss only one question on the quiz, I will give you an A.”  If you live up to your side and I give you the A, I’ve followed through on my promise, so the first line is true.  But if you miss only one question and I renege, then Q is false and the original implication must be false also. We can all agree to the first two lines of the truth table.

But when we consider the last two lines, it’s not as obvious why the statement is still true even when P is not.  If you miss more than one question, it doesn’t matter what grade I give you, I haven’t broken my promise, so the implication has NOT been proven false and is therefore still true. 

You already learned the CONVERSE, INVERSE, and CONTRAPOSITIVE of the conditional statement.  Here are the corresponding truth tables.

CONVERSE:  Q implies P.....

INVERSE:  not P implies not Q...

CONTRAPOSITIVE: not Q implies not P...

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Here’s the TRICK to accurately filling out the truth table on the next quiz.  Look at the first part of the combination. 
•If it is true and the second part is also true, the conditional statement is TRUE. 
*If the first part is true and the other part is false, the conditional statement is FALSE. 
*If the first part is false, the truth statement is TRUE, no matter what.

In other words, the only time the conditional statement is FALSE is when the first part is true but the second is false.

You probably won't find "truth tables" outside of this unit in geometry, but you'll find them useful in unexpected ways throughout life.  Need to evaluate the speeches of political candidates or sales pitches and advertisements?  Well, if the first part of the statement isn't true in the first place, then anything in the second part can be false and still not be a technical lie.  Understand truth tables and you won't be fooled in choosing your vote or buying a product or service.  This is the logic part of math and it's a valuable lesson.  In fact, LOGIC may be a course you could take in college, especially if you major in premed or prelaw.