# Chapter 1 Questions.

## Propositional Logic.

### Question 1

Which of the following propositional formulas represents the sentence, 'He will come on the 8:15 or the 9:15 train; if the former, he will have time to visit us', where

p means 'He will come on the 8:15'
q means 'He will come on the 9:15'
r means 'He will have time to visit us'

### Question 2

Which of the following sentences has the logical form (p q) r

### Question 3

Which of the following propositional formulas is satisfied by the valuation which assigns T to P, and F to q and r.

1. (p r) q
2. (r (p q))
3. r (p q)
4. p q r
5. (p q) r

6. Recall that p q is an abbreviation for (p q) (q p).

### Question 4

Which of the following propositional formulas is a tautology?
Recall that a tautology is a formula that evaluates to T for all possible assignments of truth values.
1. ( p r) (p r)
2. (p (p q))
3. r (p r)
4. p T
5. (p q) (p q)

6. Recall that p q is an abbreviation for (p q) (q p).

### Question 5

Which of the following entailments is valid?

### Question 6

Which of the columns below represents the truth table for the formula,
(p q ) (r q)?
 p q r 1 2 3 4 5 T T T F T T T T T T F T T T T F T F T F F F T T T F F T T T T T F T T F T T T F F T F T T T T F F F T F F T T T F F F T T T T T

### Question 7

Which of the following formulas represents the sentence 'If Smith has installed central heating, then he has sold his car or he has not paid his mortage', where;

p means 'Smith has installed central heating'
q means 'Smith has sold his car'
r means 'Smith has paid his mortage'.

### Question 8

Which of the following formulas represents the sentence, 'Share prices will go up, and if interest rates go up too, there will be a recession', where;

p means 'share prices will go up'
q means 'interest rates will go up'
r means 'there will be a recession'.

### Question 9

Which of the following sentences could be written a p (q r), for suitable p, q, and r ?

### Question 10

According to the standard convention about binding priorites, the formula, p q r, is implicity one of the following. Which?

### Question 11

Which of the following formulas has the parse tree:

### Question 12

Which of the following is a correct proof of,
p q r p q?

### Question 13

Which of the sequents below are valid, i.e. have a natural deduction proof?
(You do not need to state a proof.)

### Question 14

Which of the following statements about Natural Deduction is true ?

### Question 15

Which of the following are valid sequents?

### Question 16

The following is allegedly an incomplete proof of the sequent p (q r), q, r p.

 1 p (q r) premise 2 q premise 3 r premise 4 p assumption p i

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