Exercises 1

(From Löbner, pp. 82-83)

1. For each of the following sentences, say which of A or B counts as its proper negation. (Check the truth conditions of A and B: which one is necessarily true just in case the first sentence is false?)

a) It’s always raining here.
A. It’s never raining here.
B. It’s not always raining here.

b) All the kids are sick.
A. All the kids are not sick.
B. Not all the kids are sick.

c) Somebody helped me.
A. Nobody helped me.
B. Somebody didn’t help me.

d) She’s still here.
A. She’s not yet here.
B. She isn’t here anymore.

2. Which of the following statements about logical properties are true and which are false?

a) If A is logically true, then not-A is logically false.

b) If A is contingent, then not-A is either logically true or logically false.

c) A and not-A are always logically contrary.

d) If A entails not-B, then B entails not-A.

e) It is logically impossible that A entails not-A.

3. Define logical equivalence, logical contrariety, and logical contradiction in terms of logical entailment.

4. Consider the following pairs of sentences A and B. Which combinations of true and false are possible? Does one sentence entail the other? Are they equivalent? Are they contraries? contradictions? Or logically independent of each other?

a)
A. John sold the book to Mary.
B. Mary bought the book.

b)
A. I turned the light off.
B. It’s dark now.

c)
A. Many people liked the show.
B. Nobody liked the show.

d)
A. Some of the kids are sick.
B. Some of the kids are not sick.

e)
A. Only 50% of the people here have a job.
B. 50% of the people here don’t have a job.

5. Which logical relation applies to the relation between the meanings of the following pairs of words?

f) vehicle, bus

g) married, unmarried

h) buy, sell

i) bus, train

j) pleasant, unpleasant

k) above, below