Backward Reasoning – Modus Tollens Proof Problem: Election

Subsection 3.5.8 Backward Reasoning – Modus Tollens Proof Problem: Election

Election:

Assign the following names to basic statements:

C : I endorse Carol.

P : I endorse Peter.

W : Carol will win.

X : Taxes are cut.

Prove: C ∨ P I must endorse Carol or Peter.

C → W If I endorse Carol, she will win.

W → X If she wins, taxes will be cut.

¬ X Taxes cannot be cut.

∴ P I must endorse Peter.

You should do this proof yourself.

You can also watch our video, which will outline our strategy for doing this.

Exercises Exercises

Exercise Group.

1.

Prove: \(H \rightarrow ( S \vee G\) )

¬ G

Answer.

Invoke Querium ---PsQs11

Solution.

[1] H  (S  G) Premise [2] G Premise [3] H Premise [4] S  G Modus Ponens [1], [3] [5] S Disjunctive Syllogism [2], [4]

2.

Prove: ¬ p ∨ r

r → s

¬ s

¬ p

(Hint: For this one, you’re not going to be able to reason forward with Modus Ponens. You’ll want to reason “backwards”. What rule lets you do that?)

Answer.

Invoke Querium. ---PsQs12 [1] p  r Premise [2] r  s Premise [3] s Premise [4] r Modus Tollens [2], [3] [5] p Disjunctive Syllogism [1], [4]

Solution.

Explanation (to appear in Quest after return from Querium): We realized that we had to use the first premise [1], since it’s the only one that mentions p. Then we noted that, if we knew r, we could use Disjunctive Syllogism with [1] to derive p. So we used [2] and [3] to derive r and then derive our conclusion, p.

3.

3. Prove: p ∨ ¬( q ∧ r )

∴ q → ( r → p )

(Hint: You can use Conditional Disjunction to turn or s into implications.)

Answer.

Invoke Queirum. ---PsQs13 [1] p  (q  r) Premise [2] p  (q  r) De Morgan [1] [3] (q  r)  p Commutativity of or [2] [4] q  (r  p) Associativity of or [3] [5] q  (r  p) Conditional Disjunction [4] [6] q  (r  p) Conditional Disjunction [5]

Solution.

Explanation (to appear in Quest after return from Querium): This is a good example of how we can start with ands, ors, and nots and convert them into implications.

4.

4. Let’s return yet again to a famous Catch-22 situation. We’ve given names to the following statements:

C : I’m crazy.

R : I’ve requested a mental health discharge from the Army.

E : I’m eligible for a mental health discharge from the Army.

In Joseph Heller’s book, the Army has two rules about this. We have encoded them as premises as follows:

[1] E → C ∧ R Only way to be eligible is to be crazy and request the discharge.

[2] R → ¬ C I’m not crazy if I’ve requested the discharge.

We want to prove that it’s not possible that I’m eligible for a discharge. (And, since we could do this same proof for anyone else, there can never be any of these discharges.)

So prove: E → C ∧ R

R → ¬ C

∴ ¬ E

Answer.

Invoke Querium ---Catch22 [1] E → C  R Premise [2] R → C Premise [3] (C  R) → E Contrapositive [1] [4] (C  R) → E De Morgan [3] [5] R C Conditional Disjunction [2] [6] C R Commutativity [5] [7] E Modus Ponens [4], [6]