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Formal Logic
7. Justify each step in the following proof sequence of
(E x)[P(x) S Q(x)] S [(4x)P(x) S (E x)Q(x)]
1. (E x)[P(x) S Q(x)]
2. P(a) S Q(a)
3. (4x)P(x)
4. P(a)
5. Q(a)
6. (E x)Q(x)
8. Justify each step in the following proof sequence of
(E x)P(x) ` (4x)(P(x) S Q(x)) S (E x)Q(x)
1. (E x)P(x)
2. (4x)(P(x) S Q(x))
3. P(a)
4. P(a) S Q(a)
5. Q(a)
6. (E x)Q(x)
9. Consider the wff
(4x)[(E y)P(x, y) ` (E y)Q(x, y)] S (4x)(E y)[P(x, y) ` Q(x, y)]
a. Find an interpretation to prove that this wff is not valid.
b. Find the flaw in the following “proof” of this wff.
1. (4x)[(E y)P(x, y) ` (E y)Q(x, y)] hyp
2. (4x)[P(x, a) ` Q(x, a)]
1, ei
3. (4x)(E y)[P(x, y) ` Q(x, y)]
2, eg
10. Consider the wff
(4y)(E x)Q(x, y) S (E x)(4y)Q(x, y)
a. Find an interpretation to prove that this wff is not valid.
b. Find the flaw in the following “proof” of this wff.
1. (4y)(E x)Q(x, y) hyp
2. (E x)Q(x, y)
1, ui
3. Q(a, y)
2, ei
4. (4y)Q(a, y)
3, ug
5. (E x)(4y)Q(x, y) 4, eg
In Exercises 11–16, prove that each wff is a valid argument.
11. (4x)P(x) S (4x)[P(x) ~ Q(x)]
12. (4x)P(x) ` (E x)Q(x) S (E x)[P(x) ` Q(x)]
13. (E x)(E y)P(x, y) S (E y)(E x)P(x, y)
14. (4x)(4y)Q(x, y) S (4y)(4x)Q(x, y)