Topic 5: Proof Technique (Rules of Inference)

(a) If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.

• Modus Ponens

• Modus Tollens

• Hypothetical Syllogism

• invalid

If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today.

• Modus Ponens

• Modus Tollens

• Hypothetical Syllogism

• invalid

If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will get sunburned.

• Modus Ponens

• Modus Tollens

• Hypothetical Syllogism

• invalid

If I drive to work, then I will arrive tired. I am not tired when I arrive at work. I do not drive to work.

• Valid

• invalid

If I drive to work, then I will arrive tired. I do not drive to work. I will not arrive tired.

• Valid

• invalid

I will become famous or will not become a writer. I will become a writer  I will become famous.

• Valid

• invalid

If I try hard and I have talent, then I will become a musician. If I become a musician, then I will be happy. If I am not happy, then I did not try hard, or I do not have talent.

• Valid

• invalid

If my plumbing plans do not meet the construction code, then I cannot build my house. If I hire a licensed contractor, then my plumbing plans will meet the construction code. I hire a licensed contractor. I can build my house.

• Valid

• invalid

(p → q) ˄ (q → r) ~q ˄ r  p

• Valid

• invalid

~(p → q) p  ~q

• Valid

• invalid