Extra: Deduction, Induction, and Abduction

 Deduction, Induction, and Abduction

A novice's guide to logical reasoning!

Questions followed by concepts:

1. If Jane has a cat, then Jane has a pet. Jane has a cat. Therefore, Jane has a pet.

Valid
Law of detachment

2. If Jane has a cat, then Jane has a pet. Jane has a pet. Therefore, Jane has a cat.

Invalid
Converse error

3. If Jane has a cat, then Jane has a pet. It is not the case that Jane has a pet. Therefore, it is not the case that Jane has a cat.

Valid
Law of contraceptive

4. If Jane has a cat, then Jane has a pet. It is not the case that Jane has a cat. Therefore, it is not the case that Jane has a pet.

Invalid
Inverse error

5. If E.T. phones home, then blue is Joe's favourite colour. It is not the case that blue is Joe's favourite colour. Therefore, it is not the case that E.T. phones home.

Valid
Law of contrapositive

6. It is not the case that Yoda is green. If Darth Vader is Luke's dad, then Yoda is green. Therefore, it is not the case that Darth Vader is Luke's dad.

Invalid
Inverse error

7. Dan plays cello. If Mary plays the harp, then Owen plays the clarinet. Therefore, it is not the case that Mary plays the harp.

Invalid

8. All smurfs are snorks. All ewoks are snorks. Therefore, all smurfs are ewoks.

Invalid

9. Kate is a lawyer. Therefore, Kate is a lawyer.

Valid

10. If Rufus is a human being, then Rufus has the right to life. It is not the case Rufus is a human being. Therefore, it is not the case Rufus has the right to life.

Invalid
Inverse error

11. All anarchists are socialists. All socialists are totalitarians. Therefore, all anarchists are totalitarians.

Valid
Law of syllogism

12. No cat is a biped. All kangaroos are biped. Therefore, no cat is a kangaroo.

Valid
Law of detachment 

Concept:

          Converse, contrapositive, and Inverse: 

         Starting with a conditional statement: If p then q.

The converse of the conditional statement is "If q then p".

The contrapositive of the conditional statement is "If not q then not p".

The inverse of the conditional statement is "If not p then not q."

Example;" If it rained last night, then sidewalk is wet."

Converse: If the sidewalk is wet, then it rained last night.

Contrapositive: If the sidewalk is not wet, it did not rain last night.

Inverse: If it did not rain last night, then the sidewalk is not wet.

         Law of Detachment

          If p, then q, q is true as long as p.

         Use the law of detachment to derive a new true statement.

1. If you are a penguin, then you live in the Southern Hemisphere.

2. You are a penguin.

Let p be the statement you are a penguin. Let q be the statement you live in the Southern Hemisphere. Then 1 and 2 can be written: If p, then q. It is given that you are a penguin (p). Therefore, you live in the Southern Hemisphere. 

          Law of syllogism:

          It is also known as reasoning by transitivity. It is a valid argument form of deductive reasoning that follows a set pattern. It is similar to the transitive property of equality, i.e., if p=q and q=r then, p=r as in example 11.