Logical Arguments

2. 3.5 – Analyzing Arguments with Euler Diagrams Logical Arguments A logical argument is made up of premises (assumptions, laws, rules, widely held ideas, or observations) and a conclusion Valid and Invalid Arguments An argument is valid if the fact that all the premises are true forces the conclusion to be true. An argument that is not valid is invalid. It is called a fallacy. Arguments with Universal Quantifiers Several techniques can be used to check the validity of an argument. One visual technique is based on Euler Diagrams.

3. 3.5 – Analyzing Arguments with Euler Diagrams Using an Euler Diagram to Determine Validity Is the following argument valid? All cats are animals. Fuzzy is a cat. Fuzzy is an animal. x represents Fuzzy. Animals The diagram shows that Fuzzy is inside the region for “animals”. Cats x The argument is valid.

4. 3.5 – Analyzing Arguments with Euler Diagrams Using an Euler Diagram to Determine Validity Is the following argument valid? All sunny days are hot. Today is not hot Today is not sunny. x represents today Hot days x The diagram shows that today is outside the region for “sunny”. Sunny days The argument is valid.

5. 3.5 – Analyzing Arguments with Euler Diagrams Using an Euler Diagram to Determine Validity Is the following argument valid? All cars have wheels. That vehicle has wheels. That vehicle is a car. x represents “that vehicle” Things that have wheels The diagram shows “that vehicle” can be inside the region for “Cars” or outside it. x ? Cars x ? The argument is invalid.

6. 3.5 – Analyzing Arguments with Euler Diagrams Using an Euler Diagram to Determine Validity Is the following argument valid? Some students drink coffee. I am a student . I drink coffee . The diagram shows that “I” can be inside the region for “Drink coffee” or outside it. People that drink coffee I ? Students The argument is invalid. I ?

7. 3.6 – Analyzing Arguments with Truth Tables Use an Euler Diagram with quantifiers such as: “all,” “some,” “every,” “no,” etc. Use a Truth Table if the argument does not contain any quantifiers. Is the following argument valid? If the day is sunny, then it is hot. Today is not hot Today is not sunny. p: if the day is sunny p: if the day is not sunny q: it is not hot q: it is hot

8. 3.6 – Analyzing Arguments with Truth Tables p: if the day is sunny Is the following argument valid? If the day is sunny, then it is hot. Today is not hot Today is not sunny. p: if the day is not sunny q: it is hot q: it is not hot p  q q p Tautology p q q p (p  q)   q p  q ((p  q)   q)  p T T F F T F T T F F T F F T F T T F T F T F F T T T T T Valid Argument

9. 3.6 – Analyzing Arguments with Truth Tables p: I go to the fair Is the following argument valid? I will go to the fair or I will go to the beach. I did not go to the fair. I went to the beach. p: I do not go to the fair q: I go to the beach q: I do not go to the beach. p  q p q Tautology p q p (p  q)   p p  q ((p  q)   p)  q T T F T F T T F F T F T F T T T T T F F T F F T Valid Argument

10. 3.6 – Analyzing Arguments with Truth Tables p: go to the jungle Is the following argument valid? If you go to the jungle, you will see a tiger. You did not go to the jungle. You did not see a tiger. p: do not go to the jungle q: will see a tiger q: will not see a tiger p  q p q Not a tautology p q q p (p  q)   p p  q ((p  q)   p)  q T T F F T F T T F F T F F T F T T F T T F F F T T T T T Invalid Argument