Logic Text Chapter 3 Solutions

Chapter 3: Truth Tables

Basic

Question {3.1}

The asterisk is under the column of the main operator.

1:

2 and 12:

3 and 8:

4 and 14:

5 and 10:

6, 11 and 13:

7:

9:

15:

Question {3.2}

They are all tautologies except for the fourth one.

Let p = 1, q = 0 and r = 0. That makes the formula false.

Question {3.3}

1:

2:

3:

4:

The first and third propositions are equivalent to each other, and both are not equivalent to the second.

5. p & ~ p is equivalent to ~(q ⊃ q), as both are contradictions. r ∨ ~ r and s ⊃ s are equivalent as both are tautologies.

Question {3.4}

1:

The argument form is valid, as there is no row where the premises are true & the conclusion false.

2:

The argument form is valid as there is no row where the premises are both true & the conclusion false.

3:

The argument form is valid as there is no row where the premise is true & the conclusion false.

4:

The argument form is invalid as there is a row where the premises are true & the conclusion false, marked with the star.

5:

The argument is valid as there is no row where the premises are both true & the conclusion false.

6:

The argument is valid as there is no row where the premise is true & the conclusion false.

7:

The argument is valid as there is no row where the premises are true & the conclusion false. (There is no row where premises are true.)

8:

The argument is valid – there is no row where the premise is true and the conclusion false. They are true in exactly the same rows, so they are equivalent.

9:

The argument is valid, as there’s no row where the premise is true and the conclusion false. (Again, they’re equivalent.)

10:

The argument is valid. ~~p and p are equivalent, what’s more.

11:

The argument is valid – there is no row where the premise is true and the conclusion false. They are not true in exactly the same rows, so they are not equivalent.

12:

The argument is valid. There’s no row with the premise true & conclusion false.

13:

The argument is valid. There’s no row with the premise true & conclusion false.

14:

The argument is valid. There’s no row with the premise true & conclusion false.

15:

The argument is valid. There’s no row with the premise true & conclusion. In fact, the conclusion is a tautology, and any argument like this is valid.

16:

The argument is valid – there is no row where the premise is true and the conclusion false.

17:

The argument is invalid. There’s a row with the premise true & conclusion false (with the star).

18:

The argument is invalid – there is a row where the premise is true and the conclusion false. (See the asterisk.)

19:

The argument is valid – there is no row where the premise is true and the conclusion false.

20:

The argument form is valid. There’s no row where the premises are both true and the conclusion is false.

Question {3.5}

The argument form goes like this: (j & ~ b) ⊃ ~ j therefore j ⊃ b

This is a valid argument form:

Question {3.6}

The argument form goes like this: ~(b & ~ m) therefore m ⊃ b It is invalid.

Question {3.7}

The argument form is: e ⊃ c, c ⊃ p, p ⊃ ~ e therefore ~ e It is a valid form.

Question {3.8}

The argument form is: p ∨ c, p ⊃ m, c ⊃ s therefore ~(~ m &~ s) This is valid.

Question {3.9}

The argument form is: (r ⊃ ~ w) ⊃ r, therefore r. Surprisingly, this is valid. If the conclusion, r, is false, then (r ⊃ ~ w) is true, and r is false, so the premise is also false.

Question {3.10}

(j ⊃ c) & (e ⊃ d) therefore (j ⊃ d) ∨ (e ⊃ c) Surprisingly, this is valid.

Question {3.11}

Here’s a truth table for exclusive disjunction (I’ll use the symbol x for it).

The exclusive disjunction of p and q is true when exactly one of p and q is true.

Advanced:

Question {3.14}

Each formula here is a tautology. I think that this shows that ’ ⊃ ’ is not a good translation of ‘if’ in English.