Problem Set 03

Instructions

This Problem Set requires you to use both Truth Tables and Analytic Tableaux to demonstrate valid arguments and/or theorems. It is fine to write this problem set by hand, but please make sure you circle the relevant rows (in the case of proving validity) or the relevant column (in the case of proving tautologies/theorems). The problem set is due Tuesday, 9/18. Again, I do not mind students working on the problem sets in groups--it is, in fact, encouraged--but your answers must be your own. If you have any question, puzzle, or require clarification, please do not hesitate to contact me (berkich@gmail.com; 3976, 944-2756 mobile--texts strongly preferred).

A. Validity by Truth Table (5pts ea.)

1)

1. (P → (Q ∨ R))
2. ∼Q
---------------------
∴ 3. (P → R)

2)

1. ∼(P ∧ ∼Q) ∧ ∼(Q ∧ ∼P)
---------------------
∴ 2. (P ↔ Q)

3)

1. (P ↔ ∼Q)
2. (Q ↔ ∼R)
---------------------
∴ 3. (P ↔ R)

B. Tautology (theorem) by Truth Table (5pts ea.)

4) P → (P ∨ Q)

5) (P ∧ Q) → P

6) (P ∨ Q) ↔ ∼(∼Q ∧ ∼P)

C. Validity by Analytic Tableaux (10pts ea.)

7)

1. P
2. Q
---------------------
∴ 3. ∼(P → ∼Q)

8)

1. (∼P → Q)
---------------------
∴ 2. (P ∨ Q)

9)

1. (P ↔ ∼Q)
2. (Q ↔ ∼R)
---------------------
∴ 3. (P ↔ R)

D. Tautology (theorem) by Analytic Tableaux (5pts ea.)

10) (∼Q ∧ (P → Q)) → ∼P

11) (P → (Q ∧ ∼Q)) → ∼P

12) (P ∧ Q) ↔ ∼(∼P ∨ ∼Q)