Derived Rules: Modus Tollens

1. A→B
2. ~B
1. A by MT
2. ~A by MT
3. B by →out
4. Neither rule can be applied.
1. R→S
2. ~S
1. R by MT
2. ~R by MT
3. S by →out
4. Neither rule can be applied.
1. M→N
2. ~N
1. M by MT
2. ~M by MT
3. N by →out
4. Neither rule can be applied.
1. R→S
2. ~R
1. S by MT
2. ~S by MT
3. ~S by →out
4. Neither rule can be applied.
1. R→S
2. R
1. R by MT
2. ~R by MT
3. S by →out
4. Neither rule can be applied.
1. B→D
2. ~B
1. D by MT
2. ~D by MT
3. ~D by →out
4. Neither rule can be applied.
1. M→N
2. N
1. M by MT
2. ~M by MT
3. M by →out
4. Neither rule can be applied.
1. A→B
2. B
1. A by MT
2. ~A by MT
3. A by →out
4. Neither rule can be applied.
1. S→W
2. S
1. W by MT
2. ~W by MT
3. W by →out
4. Neither rule can be applied.
1. T→W
2. ~T
1. W by MT
2. ~W by MT
3. ~W by →out
4. Neither rule can be applied.
1. ~R→T
2. ~T
1. R by MT
2. ~R by MT
3. ~~R by MT
4. No rule can be applied.
1. ~M→~N
2. ~~M
1. N by MT
2. ~N by MT
3. ~~N by →out
4. ~N by →out
5. No rule can be applied.
1. ~A→B
2. ~B
1. ~A by MT
2. A by MT
3. ~~A by MT
4. ~A by →out
5. No rule can be applied.
1. ~R→~S
2. ~R
1. S by MT
2. ~S by MT
3. ~~S by →out
4. ~S by →out
5. No rule can be applied.
1. ~R→~S
2. R
1. S by MT
2. ~S by MT
3. S by →out
4. ~S by →out
5. No rule can be applied.
1. ~A→~B
2. ~~A
1. ~B by →out
2. ~~B by →out
3. ~B by MT
4. ~~B by MT
5. No rule can be applied.
1. ~N→~M
2. ~M
1. ~N by MT
2. N by MT
3. ~~N by MT
4. ~N by →out
5. No rule can be applied.
1. ~D→~E
2. ~~E
1. ~D by MT
2. D by MT
3. ~~D by MT
4. ~D by →out
5. No rule can be applied.
1. ~~D→H
2. ~H
1. ~~~D by MT
2. ~~D by MT
3. ~D by MT
4. ~~D by →out
5. No rule can be applied.
1. ~~R→~T
2. ~~T
1. ~R by MT
2. ~~~R by MT
3. ~~R by MT
4. ~R by →out
5. No rule can be applied.
1. R→~S
2. S
1. R by MT
2. ~R by MT
3. R by →out
4. ~R by →out
5. No rule can be applied.
1. ~~A→B
2. A
1. B by MT
2. ~B by MT
3. B by →out
4. ~B by →out
5. No rule can be applied.
1. ~(A&B)→C
2. ~C
1. ~(A&B) by MT
2. ~~(A&B) by MT
3. (A&B) by MT
4. ~(A&B) by →out
5. No rule can be applied.
1. (A∨B)→C
2. ~C
1. ~(A∨B) by MT
2. ~~(A∨B) by MT
3. (A∨B) by MT
4. (A∨B) by →out
5. No rule can be applied.
1. (R→S)→~T
2. ~~T
1. ~(R→S) by MT
2. ~~(R→S) by MT
3. (R→S) by MT
4. (R→S) by →out
5. No rule can be applied.
1. ~(D→E)→~E
2. ~~E
1. ~(D→E) by MT
2. (D→E) by MT
3. ~~(D→E) by MT
4. ~(D→E) by →out
5. No rule can be applied.
1. ~(R&S)→(T&W)
2. ~(T&W)
1. ~(R&S) by MT
2. ~~(R&S) by MT
3. (R&S) by MT
4. (R&S) by →out
5. No rule can be applied.
1. ~~(M∨N)→(D&~E)
2. ~~(D&~E)
1. ~~(M∨N) by MT
2. (M∨N) by MT
3. ~(M∨N) by MT
4. ~~~(M∨N) by →out
5. No rule can be applied.
1. (R→S)→(~A→B)
2. ~(~A→B)
1. (R→S) by MT
2. ~~(R→S) by MT
3. ~(R→S) by MT
4. ~(R→S) by →out
5. No rule can be applied.
1. ~~~(M∨N)→~(C&D)
2. ~~~(M∨N)
1. ~~(C&D) by MT
2. (C&D) by MT
3. ~(C&D) by MT
4. ~(C&D) by →out
5. No rule can be applied.
1. ~~(A→B)→~(~~M→~N)
2. ~(~~M→~N)
1. ~~(A→B) by MT
2. (A→B) by MT
3. ~~~(A→B) by MT
4. ~~(A→B) by →out
5. No rule can be applied.
1. ~~(~R&~S)→~(~T&~~W)
2. ~~(~T&~~W)
1. ~~~(~R&~S) by MT
2. ~~(~R&~S) by MT
3. ~(~R&~S) by MT
4. (~R&~S) by →out
5. No rule can be applied.
1. ~(B→(C→~D))→~(~(M→~N)→~~O)
2. ~~(~(M→~N)→~~O)
1. (B→(C→~D)) by →out
2. ~(B→(C→~D)) by MT
3. ~~(B→(C→~D)) by MT
4. ~(B→(C→~D)) by →out
5. No rule can be applied.

Derived Rules: Modus Tollens

This exercise focuses on Modus Tollens (MT) but a few instances of →out are included because students sometimes confuse these rules. For each question decide whether MT or →out may be applied, and choose the correct answer.