Use truth-tables to determine which of the following are tautological, which are contradictory, and which are contingent.
1. ~ (A ⊃ A)
2. A ⊃ ~ A
3. A ≡ (A v B)
4. (A & ~A) ⊃ B
5. (A & B) ⊃ ~ (~A v ~B)
Use truth-tables to determine which of the following arguments below are valid and which are invalid. In each problem, the premises are listed first and the conclusion follows the slash.
1. A v B
~A / ~B
2. A ⊃ B / ~A ⊃ ~B
3. A ⊃ B
B ⊃ C / A ⊃ C
4. A ⊃ B
A / B
5. ~A v ~B / ~ (A & B)
Use partial truth-tables to prove that each of the following symbolized arguments is invalid. In each case, the argument’s symbolized premises are written first and the symbolized conclusion appears after the slash mark.
1. A v B
A / ~B
2. A ⊃ B
B ⊃ W
W ⊃ S
S ⊃ H / H ⊃ A
3. ~(A & B) / ~A & ~B
4. A ⊃ B
W ⊃ B / A ⊃ W
5. A ⊃ B
B / A
Use truth-tables to determine, for each pair of sentences below, whether the sentences are equivalent or not equivalent. Note that a comma separates the members of each pair.
1. ~ ~P , P
2. ~(P & Q) , ~ P v ~Q
3. P ⊃ Q , Q ⊃ P
4. P ≡ Q , [(P ⊃ Q) & (Q ⊃ P)]
5. (P v Q) v R , P v (Q v R)
In each proof below, the conclusion is the formula that follows the slanted slash mark and the premises are the numbered formulas that precede the conclusion. These proofs are complete except that the justifications for the derived lines have not been filled in. Using the first four rules, supply a justification for each derived line by filling in the appropriate rules and line numbers.
* The first four rules are:
The disjunctive syllogism rule (DS)
The modus ponens rule (MP)
The modus tollens rule (MT)
The hypothetical syllogism rule (HS)
(1):
1. J v ~ (I v R)
2. S ⊃ ~ J
3. H v S
4. ~H
5. ~(I v R) ⊃ A / A
6. S
7. ~ J
8. ~(I v R)
9. A
(2):
1. J ⊃ (S & I)
2. ~ J ⊃ (B v R)
3. ~ (S & I)
4. (B v R) ⊃ (I v H) / I v H
5. ~ J
6. B v R
7. I v H
(3):
1. S v ~ (H & S)
2. S ⊃ G
3. J ⊃ ~ G
4. M ⊃ (H & S)
5. J / ~M
6. ~ G
7. ~ S
8. ~ (H & S)
9. ~ M
(4):
1. (H & G) ⊃ (F ⊃ R)
2. J & I
3. A v ~ (F ⊃ R)
4. ~A / ~ (H & G)
5. ~ (F ⊃ R)
6. ~ (H & G)
(5):
1. A ⊃ B
2. B ⊃ R
3. (A ⊃ R) ⊃ G / G
4. A ⊃ R
5. G
(6):
1. A v E
2. ~ A
3. E ⊃ [~ A ⊃ (E ⊃ S)] / S
4. E
5. ~ A ⊃ (E ⊃ S)
6. E ⊃ S
7. S
Each of the following symbolized arguments is valid. Using the first four rules, provide a proof for each argument.
(1):
1. (E ≡ F) v ~ (A & B)
2. H ⊃ (A & B)
3. ~(E ≡ F) / ~ H
(2):
1. F ⊃ S
2. S ⊃ G
3. (F ⊃ G) ⊃ M / M
(3):
1. ~ F ⊃ ~ S
2. ~ S ⊃ ~ G
3. (~F ⊃ ~ G) ⊃ H / H
(4):
1. A ⊃ B
2. B ⊃ E
3. B ⊃ R
4. ~ R
5. A v B / E