Valid and Invalid Arguments

Tom Kelliher, MA 115

Sept. 17, 1997

Announcements

Quiz on Friday:

1. 10--15 minutes.

2. One problem from Section 1.1 and one problem from Section 1.2.

3. You may bring one two-sided sheet of notes.

Assignment

1. The following problems from Section 1.3 are due Monday: 11, 28, 29, 37.b.

2. Friday, we will discuss material from Sections 1.4 and 1.5, but you will not be responsible for it.

Valid Arguments

1. What did we say constituted a valid argument?

2. Can one mechanically check that an argument form is valid?

3. What is an argument form?

Modus Ponens

Method by affirming:

Remember: p is a sufficient condition.

Modus Tollens

Method by denying:

Use the contrapositive to see similarity to Modus Ponens.

Disjunctive Addition

Generalization:

Would it be valid to go the other way?

Conjunctive Simplification

Simplification:

Conjunctive Addition

Disjunctive Syllogism

Process of elimination.

Hypothetical Syllogism

Remind anyone of transitivity?

Dilemma: Proof by Division into Cases

Can this be generalized to more cases?

Generalize to three cases:

Rule of Contradiction

where c is a contradiction.

Fallacious Arguments

How can we prove the previous forms, again?

Converse Error

Not!

Inverse Error

Again, Not!

Two Examples

1. The famous detective Percule Hoirot was called in to solve a baffling murder mystery. He determined the following facts:
   1. Lord Hazelton, the murdered man, was killed by a blow on the head with a brass candlestick.

   2. Either Lady Hazelton or a maid, Sara, was in the dining room at the time of the murder.

   3. If the cook was in the kitchen at the time of the murder, then the butler killed Lord Hazelton with a fatal dose of strychnine.

   4. If Lady Hazelton was in the dining room at the time of the murder, then the chauffeur killed Lord Hazelton.

   5. If the cook was not in the kitchen at the time of the murder, then Sara was not in the dining room when the murder was committed.

   6. If Sara was in the dining room at the time the murder was committed, then the wine steward killed Lord Hazelton.

   Who was the murderer?

2. Sharky, a leader of the underworld, was killed by one of his own band of four henchmen. Detective Sharp interviewed the men and determined that all were lying except for one. He deduced who killed Sharky on the basis of the following statements:
   1. Socko: Lefty killed Sharky.

   2. Fats: Muscles didn't kill Sharky.

   3. Lefty: Muscles was shooting craps with Socko when Sharky was knocked off.

   4. Muscles: Lefty didn't kill Sharky.

   Who killed Sharky?

In-Class Exercises

Work in groups of two on the following problems from the end of Section 1.3: 3, 6, 35, 36.