A Brief and Fundamental Summary of Logic and Reasoning
Logic is the study of the principles or rules for valid and consistent reasoning. (Webster’s Dictionary)
Reasoning is the capacity for rational thought, inference, or discrimination. (Webster’s Dictionary)
Types
of Reasoning:
Inductive Reasoning: ( Specific to General )
Many specific examples leading to a general conclusion.
Example: Seeing a few examples a person draws a general conclusion about students.
Deductive Reasoning: ( General to Specific )
A general statement leads to conclusions about many examples.
Example: A commercial on TV leads people to believe it will happen in all cases.
Types of Deductive Statements:
If ( Given Information ) Then ( Statement to be Proved )
Hypothesis Conclusion
Conditional: |
If A = B |
Then A+C = B+C |
Converse |
If A+C = B+C |
Then A = B |
Inverse: |
If A <> B |
Then A+C <> B+C |
Contrapositive: |
If A+C <> B+C |
Then A <> B |
Rectangle of Reason shows proving only two of the four statements thus proves all.
Converse |
Conditional |
Contrapositive |
Inverse |
Types
of Logical Situations:
Dichotomy Situation: ( Only Two Possibilities )
Equal or Not Equal True or False
Guilty or Innocent Male or Female
Trichotomy Situation: ( Only Three Possibilities )
Greater Than Equal Less Than
Always Sometimes Never
True Maybe False
Guilty Innocent No Contest
Types of Proofs: ( Definition of Proof: When you convince someone what you say is true. )
Direct Proof: Proves the Original Statement to be True. )
Proof by Testimonial Proof by Analogy
Proof by Induction Proof by Deduction
( Many TV Commercials use these types of proofs to convince you to buy their product.)
Indirect Proof: Proves the Alternate Statement to be False.
Mathematical Induction of Infinite sequences is an example of Indirect Proof.
Many trial lawyers try to prove Guilty is not possible thus Innocent is true.
( Many persons try to prove using examples or cases thus leading to a general conclusion. )
Reference: The Teaching of Mathematics from Counting
to Calculus, Harold P. Fawcett and Kenneth B. Cummins