7 Seventh Week

Question 1

What is the difference between “strict” and “loose” universal generalizations?

Answer 1

A strict universal generalization asserts that something is true in every single case without even one exception. (Example: Every odd number has an odd square.)

A loose universal generalization asserts that something is true in most cases, or in typical cases. (Example: Cats like roast chicken.)

Question 2

What is the difference between a “domain of quantification” and a “universe of discourse”?

Answer 2

There is no difference – these two phrases are synonyms.

Question 3

What is a domain of quantification?

Answer 3

When one says “everything”, it is rare that one means to consider every single thing in the whole universe without restriction. Usually one means to include only the things within some restricted category – this restricted category is the “domain of quantification”.

For example, if I’m at a party and I say “Everyone looks so glamorous!”, I don’t mean that everyone in the world looks glamorous, I mean that everyone at the party looks glamorous. The domain, in this case, is people at the party.

Question 4

What is a vacuous universal generalization? Are vacuous universal generalizations true or false?

Answer 4

The statement “Every A is a B” is said to be vacuous if there are no As. Logicians consider all vacuous universal generalizations to be true.

Question 5

Draw a Venn diagram for ‘No dog is black’.

Answer 5

Question 6

Draw a Venn diagram for ‘Every dog is black’.

Answer 6

Question 7

Draw a Venn diagram for ‘Some dog is black’.

Answer 7

Question 8

Draw a Venn diagram for ‘There is a dog that isn’t black’.

Answer 8

Question 9

What does this diagram mean in words?

Answer 9

Something is black.

Question 10

What are A, E, I and O statements?

Answer 10

Question 11

In our logical symbolism, which letters are used as variables?

Answer 11

We usually use letters from the end of the alphabet as variables: x, y, z.

Question 12

Symbolize ‘Every whale is a mammal’.

Answer 12

∀x(Wx → Mx)

Question 13

Symbolize ‘No whale is a reptile’.

Answer 13

∀x(Wx → ~Rx)

Question 14

Write down an instance of ∀x(Wx → Mx).

Answer 14

(Wa → Ma)

(You don’t have to use ‘a’ here – any name is fine.)

Question 15

Symbolize
(a) Every snob is an opera lover.
(b) Only snobs are opera lovers.

Answer 15

(a) ∀x(Sx → Ox)
(b) ∀x(Ox → Sx)