Question {3.1}
The asterisk is under the column of the main operator.
1:
y | ~ | y | |
0 | 1 | 0 | |
1 | 0 | 1 | |
* |
2 and 12:
y | p | ~ | y | ∨ | p | y | ≡ | p | ||
0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ||
0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | ||
1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | ||
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | ||
* | * |
3 and 8:
y | c | ~ | y | ≡ | c | ~ | (y | ≡ | c) | ||
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | ||
0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | ||
1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | ||
1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | ||
* | * |
4 and 14:
y | c | p | y | ⊃ | (c | ⊃ | p) | (y | ⊃ | c) | ⊃ | p | ||
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | ||
0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | ||
0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | ||
0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | ||
1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ||
1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | ||
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
* | * |
5 and 10:
y | c | p | (y | & | ~ | p) | ⊃ | c | y | ≡ | (c | ∨ | ~ | p) | ||
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | ||
0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ||
0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | ||
0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | ||
1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ||
1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | ||
1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | ||
1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | ||
* | * |
6, 11 and 13:
y | c | y | & | c | y | ∨ | c | y | ⊃ | c | |||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | |||
0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | |||
1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||
* | * | * |
7:
y | p | ~ | (y | & | p) | |
0 | 0 | 1 | 0 | 0 | 0 | |
0 | 1 | 1 | 0 | 0 | 1 | |
1 | 0 | 1 | 1 | 0 | 0 | |
1 | 1 | 0 | 1 | 1 | 1 | |
* |
9:
c | ~ | ~ | c | |
0 | 0 | 1 | 0 | |
1 | 1 | 0 | 1 | |
* |
15:
y | c | (y | ⊃ | c) | ≡ | (c | ⊃ | y) | |
0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | |
0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
* |
Question {3.2}
They are all tautologies except for the fourth one.
((p | & | q) | ⊃ | r) | ⊃ | (p | ⊃ | r) |
1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
Let p = 1, q = 0 and r = 0. That makes the formula false.
Question {3.3}
1:
p | ~ | ~ | p | |
0 | 0 | 1 | 0 | |
1 | 1 | 0 | 1 |
2:
p | q | p | & | q | q | & | p | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | ||
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3:
p | q | p | ⊃ | q | ~ | p | ∨ | q | ||
0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | ||
0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ||
1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | ||
1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |
4:
p | q | r | p | ⊃ | (q | ⊃ | r) | (p | ⊃ | q) | ⊃ | r | ~ | p | ∨ | ~ | (q | & | ~ | r) | |||
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | |||
0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | |||
0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | |||
0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | |||
1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | |||
1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | |||
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
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:
p | q | p | p | ⊃ | q | q | |||
0 | 02 | 0 | 0 | 1 | 0 | 0 | |||
0 | 1 | 0 | 0 | 1 | 1 | 1 | |||
1 | 0 | 1 | 1 | 0 | 0 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument form is valid, as there is no row where the premises are true & the conclusion false.
2:
p | q | p | q | p | ≡ | q | |||
0 | 0 | 0 | 0 | 0 | 1 | 0 | |||
0 | 1 | 0 | 1 | 0 | 0 | 1 | |||
1 | 0 | 1 | 0 | 1 | 0 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument form is valid as there is no row where the premises are both true & the conclusion false.
3:
p | q | p | & | q | p | ≡ | q | ||
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | ||
0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | ||
1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument form is valid as there is no row where the premise is true & the conclusion false.
4:
p | q | p | q | ⊃ | p | q | ||||
0 | 0 | 0 | 0 | 1 | 0 | 0 | ||||
0 | 1 | 0 | 1 | 0 | 0 | 1 | ||||
1 | 0 | 1 | 0 | 1 | 1 | 0 | * | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument form is invalid as there is a row where the premises are true & the conclusion false, marked with the star.
5:
p | q | p | q | p | & | q | |||
0 | 0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 1 | 0 | 1 | 0 | 0 | 1 | |||
1 | 0 | 1 | 0 | 1 | 0 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument is valid as there is no row where the premises are both true & the conclusion false.
6:
p | q | p | p | ∨ | q | ||
0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 1 | 0 | 0 | 1 | 1 | ||
1 | 0 | 1 | 1 | 1 | 0 | ||
1 | 1 | 1 | 1 | 1 | 1 |
The argument is valid as there is no row where the premise is true & the conclusion false.
7:
p | q | p | ≡ | q | p | ≡ | ~ | q | ~ | p | |||
0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | |||
0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | |||
1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | |||
1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
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:
p | q | r | p | ⊃ | (q | ⊃ | r) | q | ⊃ | (p | ⊃ | r) | ||
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | ||
0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | ||
0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | ||
0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | ||
1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | ||
1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | ||
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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:
p | p | ⊃ | ~ | p | ~ | p | ||
0 | 0 | 1 | 1 | 0 | 1 | 0 | ||
1 | 1 | 0 | 0 | 1 | 0 | 1 |
The argument is valid, as there’s no row where the premise is true and the conclusion false. (Again, they’re equivalent.)
10:
p | ~ | ~ | p | p | ||
0 | 0 | 1 | 0 | 0 | ||
1 | 1 | 0 | 1 | 1 |
The argument is valid. ~~p and p are equivalent, what’s more.
11:
p | q | r | p | ⊃ | q | (r | ⊃ | p) | ⊃ | (r | ⊃ | q) | ||
0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | ||
0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | ||
0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | ||
0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | ||
1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | ||
1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | ||
1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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:
p | q | p | ⊃ | q | ~ | q | ⊃ | ~ | p | ||
0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | ||
0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | ||
1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ||
1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
The argument is valid. There’s no row with the premise true & conclusion false.
13:
p | q | p | ~ | p | ⊃ | q | ||
0 | 0 | 0 | 1 | 0 | 0 | 0 | ||
0 | 1 | 0 | 1 | 0 | 1 | 1 | ||
1 | 0 | 1 | 0 | 1 | 1 | 0 | ||
1 | 1 | 1 | 0 | 1 | 1 | 1 |
The argument is valid. There’s no row with the premise true & conclusion false.
14:
p | q | p | ⊃ | (p | ⊃ | q) | p | ⊃ | q | ||
0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ||
0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | ||
1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument is valid. There’s no row with the premise true & conclusion false.
15:
p | q | p | q | ⊃ | q | ||
0 | 0 | 0 | 0 | 1 | 0 | ||
0 | 1 | 0 | 1 | 1 | 1 | ||
1 | 0 | 1 | 0 | 1 | 0 | ||
1 | 1 | 1 | 1 | 1 | 1 |
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:
p | q | r | p | ∨ | q | ~ | q | ∨ | r | p | ∨ | r | |||
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | |||
0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | |||
0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | |||
0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | |||
1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | |||
1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | |||
1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument is valid – there is no row where the premise is true and the conclusion false.
17:
p | q | p | ⊃ | q | q | ⊃ | p | |||
0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | |||
0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | * | ||
1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument is invalid. There’s a row with the premise true & conclusion false (with the star).
18:
p | q | r | p | ⊃ | (q | ⊃ | r) | (p | ⊃ | q) | ⊃ | r | |||
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | |||
0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | |||
0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | * | ||
0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | |||
1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | |||
1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | |||
1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The argument is invalid – there is a row where the premise is true and the conclusion false. (See the asterisk.)
19:
p | q | r | (p | & | q) | ⊃ | r | p | ⊃ | (~ | q | ∨ | r) | ||
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | ||
0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | ||
0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | ||
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | ||
1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | ||
1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | ||
1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |
The argument is valid – there is no row where the premise is true and the conclusion false.
20:
p | q | r | s | p | ⊃ | q | r | ⊃ | s | (q | ⊃ | r) | ⊃ | (p | ⊃ | s) | |||
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | |||
0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | |||
0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | |||
0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | |||
0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | |||
0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | |||
0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | |||
0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | |||
1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | |||
1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | |||
1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | |||
1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | |||
1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | |||
1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | |||
1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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:
j | b | (j | & | ~ | b) | ⊃ | ~ | j | j | ⊃ | b | ||
0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ||
0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | ||
1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | ||
1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
Question {3.6}
The argument form goes like this: ~(b & ~ m) therefore m ⊃ b It is invalid.
~ | (b | & | ~ | m) | m | ⊃ | b | |
1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
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).
p | q | p | x | q | |
0 | 0 | 0 | 0 | 0 | |
0 | 1 | 0 | 1 | 1 | |
1 | 0 | 1 | 1 | 0 | |
1 | 1 | 1 | 0 | 1 |
The exclusive disjunction of p and q is true when exactly one of p and q is true.
Question {3.14}
Each formula here is a tautology. I think that this shows that ’ ⊃ ’ is not a good translation of ‘if’ in English.