8.E: Chapter Seven (Exercises)
Identify which (single) rule is being exemplified in each inference. Remember that the underline separates premises from their conclusion.
A. \(\begin{align*} & [(A \wedge \neg B) \rightarrow (P \vee Q)] \\ & \overline{[ \neg (P \vee Q) \rightarrow \neg (A \wedge \neg B)]} \end{align*}\)
B. \(\begin{align*} & [(P \vee Q) \vee \neg (A \wedge \neg B)] \\ & \underline{\neg \neg (A \wedge \neg B) \ \ } \\ & (P \vee Q) \end{align*}\)
C. \(\begin{align*} & \underline{[((Z \vee X) \rightarrow Y) \wedge (Y \leftrightarrow W)]} \\ & ((Z \vee X) \rightarrow Y) \end{align*}\)
D. \(\begin{align*} & [((Z \vee X) \rightarrow Y) \rightarrow (Y \leftrightarrow W)] \\ & \underline{((Z \vee X) \rightarrow Y) \ } \\ & (Y \leftrightarrow W) \end{align*}\)
E. \(\begin{align*} & [ \neg B \rightarrow (P \vee Q)] \\ & \underline{\neg (P \vee Q) \ \ } \\ & \neg \neg B \end{align*}\)
F. \(\begin{align*} & [((Z \vee X) \rightarrow Y) \vee (Y \leftrightarrow W)] \\ & \underline{[(((Z \vee X) \rightarrow Y) \rightarrow T) \wedge ((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z))]} \\ & [T \vee \neg (W \leftrightarrow Z)] \end{align*}\)
G. \(\begin{align*} & \underline{(((Z \vee X) \vee Y) \vee T)} \\ & (Z \vee (X \vee Y)) \vee T) \end{align*}\)
H. \(\begin{align*} & \underline{[ \neg (P \vee Q) \vee (A \leftrightarrow \neg B)]} \\ & [(P \vee Q) \rightarrow (A \leftrightarrow \neg B)] \end{align*}\)
I. \(\begin{align*} & \underline{[(P \vee Q) \rightarrow (A \leftrightarrow \neg B)]} \\ & [ \neg (P \vee Q) \vee (A \leftrightarrow \neg B)] \end{align*}\)
J. \(\begin{align*} & \underline{(((Z \vee X) \rightarrow Y) \rightarrow T) \ } \\ & (((Z \vee X) \rightarrow Y) \rightarrow T) \vee [((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z))] \end{align*}\)
K. \(\begin{align*} & [((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z)) \rightarrow (T \vee \neg (W \leftrightarrow Z))] \\ & \underline{((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z)) \ } \\ & (T \vee \neg (W \leftrightarrow Z)) \end{align*}\)
L. \(\begin{align*} & (((Z \vee X) \rightarrow Y) \rightarrow T) \\ & (T \rightarrow \neg (W \leftrightarrow Z)) \\ & \overline{[((Z \vee X) \rightarrow Y) \rightarrow \neg (W \leftrightarrow Z)]} \end{align*}\)
Derive the conclusion (after the slash) from the numbered premises using only the first 4 rules we learned in section 7.2.
A.
- 1. ~C\(\supset\) (A\(\supset\)C)
- 2. ~C /~A
B.
- 1. ~W\(\supset\) [~W\(\supset\) (X\(\supset\)W)]
- 2. ~W / ~X
C.
- 1. (P \(\rightarrow\) Q)
- 2. (Q \(\rightarrow\) R)
- 3. (R \(\rightarrow\) S)
- 4. (S \(\rightarrow\) T) / (P \(\rightarrow\) T)
D.
- 1. (((Z \(\vee\) X) \(\rightarrow\) Y) \(\rightarrow\) T)
- 2. (T \(\rightarrow\) \(\neg\) (W \(\leftrightarrow\) Z))
- 3. ((Z \(\vee\) X) \(\rightarrow\) W)
- 4. (W \(\rightarrow\) Y) / \(\neg\) (W \(\leftrightarrow\) Z
E.
- 1. ~M \(\vee\) (B \(\vee\) ~T)
- 2. (B \(\supset\) W)
- 3. ~~M
- 4. ~W / ~T
F.
- 1. (~S \(\supset\) D)
- 2. [~S ∨ (~D \(\supset\) K)]
- 3. ~D / K
G.
- 1. [A \(\supset\) (E \(\supset\) ~F)]
- 2. [H \(\vee\) (~F \(\supset\) M)]
- 3. A
- 4. ~H / (E \(\supset\) M)
H.
- 1. [G \(\supset\) [~O \(\supset\) (G \(\supset\) D)]]
- 2. (O \(\vee\) G)
- 3. ~O / D
I.
- 1. P\(\supset\) (G\(\supset\)T)
- 2. Q\(\supset\) (T\(\supset\)E)
- 3. P
- 4. Q / (G\(\supset\)E)
J.
- 1. (~S \(\supset\) D)
- 2. ~S \(\vee\) (~D\(\supset\) K)
- 3. ~D /K
K.
- 1. X \(\rightarrow\) (Y \(\rightarrow\) Z)
- 2. X \(\rightarrow\) (Z \(\rightarrow\) W)
- 3. (T \(\vee\) X)
- 4. ~T / (Y \(\rightarrow\) W)
L.
- 1. X \(\vee\) (Y \(\wedge\) Z)
- 2. (Y \(\wedge\) Z) \(\rightarrow\) W
- 3. (X \(\rightarrow\) T)
- 4. \(\neg\) T /W
Derive the conclusion (after the slash) from the numbered premises using only the first 8 rules (the rules of implication) we learned in sections 7.2 and 7.3.
A.
- 1. (~A \(\rightarrow\) H)
- 2. (R \(\rightarrow\) ~B)
- 3. (~A \(\vee\) R) / (H \(\vee\) ~B)
B.
- 1. E \(\supset\) (A \(\bullet\) C)
- 2. A \(\supset\) (F \(\bullet\) E)
- 3. E / F
C.
- 1. (~F \(\vee\) M) \(\supset\) (P \(\vee\) B)
- 2. (F \(\supset\) P)
- 3. ~P / B
D.
- 1. M \(\supset\) (F\(\bullet\)G)
- 2. (F \(\supset\) K)
- 3. W
- 4. (W \(\supset\) M) / K
E.
- 1. (M\(\supset\)F) \(\bullet\) (Z\(\supset\)W)
- 2. (K \(\bullet\) L) \(\bullet\) A
- 3. K\(\supset\) (M \(\vee\) Z) / (F \(\vee\) W)
F.
- 1. (M\(\supset\)F) \(\bullet\) L
- 2. (F\(\supset\)G) \(\bullet\) A
- 3. (M\(\supset\)G) \(\supset\) [(M\(\supset\)F) \(\supset\)W] / W
G.
- 1. (M \(\bullet\) F) \(\vee\) (G \(\bullet\) W)
- 2. (M \(\bullet\) F) \(\supset\)L
- 3. (~L \(\bullet\) A)
- 4. G \(\supset\) (N \(\bullet\) O) / N
H.
- 1. (F \(\bullet\) A) \(\supset\) (G \(\bullet\) K)
- 2. (M\(\supset\)F) \(\bullet\) G
- 3. (M \(\bullet\) L)
- 4. (M\(\supset\)F) \(\supset\) A / (G \(\vee\) W)
I.
- 1. (M \(\supset\) P)
- 2. (M \(\bullet\) R)
- 3. P \(\supset\) (Q \(\bullet\) S)
- 4. (P\(\bullet\)Q) \(\supset\) (P\(\equiv\)Q) / (P\(\equiv\)Q)
J.
- 1. A \(\supset\) (B\(\supset\)C)
- 2. (A \(\bullet\) M)
- 3. B \(\bullet\) (F\(\bullet\)G) / (C \(\bullet\) A)
Derive the conclusion (after the slash) from the numbered premises using any rule we’ve learned in this text.
A.
- 1. (X \(\supset\) ~B)
- 2. (D \(\vee\) X)
- 3. B /D
B.
- 1. [Q \(\vee\) (A \(\vee\) C)]
- 2. \(\neg\) C / (A \(\vee\) Q)
C.
- 1. ( \(\neg\) M\(\supset\)B) \(\bullet\) ( \(\neg\) N\(\supset\)Q)
- 2. \(\neg\) (M \(\bullet\) N) / [(B \(\vee\) Q) \(\vee\) \(\neg\) Z]
D.
- 1. (~X \(\vee\) Y)
- 2. (~Y \(\vee\) Z)
- 3. (X \(\bullet\) W) /(X\(\bullet\)Y) \(\bullet\) Z
E.
- 1. (~X \(\supset\) T)
- 2. (W \(\bullet\) ~T)
- 3. (X \(\vee\) Y) \(\supset\) Z / (W\(\bullet\)Z)
F.
- 1. (X \(\bullet\) ~Z)
- 2. (Y \(\vee\) X) \(\supset\) ~W /~(Z \(\vee\) W)
G.
- 1. ~(X \(\vee\) ~X) / Y
H.
- 1. (G \(\supset\) E)
- 2. (H \(\supset\) ~E) / (G \(\supset\) ~H)
I.
- 1. (~N \(\vee\) B)
- 2. (N \(\supset\) B) \(\supset\) T / T
J.
- 1. ~X
- 2. ~(B \(\bullet\) Q)
- 3. (~X \(\supset\) B) / ~Q
K.
- 1. (Y \(\bullet\) Z) \(\supset\) X
- 2. (~X \(\bullet\) ~Z)
- 3. Y \(\supset\) (Y \(\bullet\) Z) / (Z \(\vee\) ~Y)
L.
- 1. (F \(\supset\) ~J)
- 2. H \(\supset\) (F \(\vee\) G)
- 3. (G \(\supset\) ~K)
- 4. H / ~(J\(\bullet\)K) \(\bullet\) (H \(\vee\) F)
M.
- 1. (A \(\supset\) ~~B)
- 2. A \(\vee\) (~C \(\vee\) ~E)
- 3. ~B
- 4. D \(\supset\) (C \(\bullet\) E) / ~D
N.
- 1. (A \(\vee\) M)
- 2. (A \(\supset\) M) \(\bullet\) (A \(\supset\) O)
- 3. (M \(\supset\) O)
- 4. (M \(\supset\) B) \(\bullet\) (O \(\supset\) A) / (O \(\vee\) B)