Types of Reasoning

Deductive Reasoning Inductive Reasoning

Both play an important role in interpreting and drawing conclusions from data analysis

Deductive Reasoning

Goes from the general to the specific

Also known as top-down logic

Seeks to prove statements of the form “If A, then B”

Such reasoning always implies three underlying components: assumptions (“If A”), methods of proof (“then”), and conclusions (“B”)

The purest applications of deductive reasoning are in the field of mathematics

Two of the most used approaches are direct proofs and transposition

Direct proofs

Proof that begins with assumptions, explains methods of proof, and states the conclusion(s)

Transposition

Any time a group of assumptions implies a conclusion, then it is also true that any time the conclusion does not hold, at least one of the assumptions must not hold

Let’s prove the following statement by direct proof:

If X and Y are odd numbers, then their sum (X + Y) is an even number

An Example:

If X = 5(odd) and Y = 9(odd), then their sum X + Y = 14 is an even number

Failing to find a contradiction is not the same a proving a statement is generally true

Direct Proof: A Mathematical Approach

If X and Y are odd numbers, then their sum (X + Y) is an even number

X and Y are odd numbers

If X is an odd number, then X can be written as X=2K+1, where K is an integer. (Example: X=13 X=(2 × 6)+1)

If Y is an odd number, then Y can be written as Y =2M+1, where M is an integer, (Example: Y=23 Y=(2 × 11)+1)

X+Y=(2K+1)+(2M+1)=2K+2M+2=2(K+M+1)

K+M+1 is an integer so X+Y is 2 times an integer

Any number that is 2 times an integer is divisible by 2

This means X+Y is even

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