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

 

Tom Love                       Malone College                       Fall 2003