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Subsection 3.6.4 Contradictory Premises and Conditionalization Proof Problem

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Prove: ยฌ( q โˆง s )

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q โˆจ p

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ยฌ p

  • s โ†’ r

Notice that youโ€™ve seen this problem before. This time, use Conditionalization to complete your proof.

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

Video cover image

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Conditionalization Problems

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Exercises Exercises

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Exercise Group.

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1.

1. Prove: ( A โˆง (ยฌ B โˆง ยฌ C )) โ†’ ( A โˆจ ยฌ( B โˆง C ))

(Hint: Think about using Simplification and/or Addition to remove terms from a conjunction or add terms to a disjunction.)

Answer.
Solution.
Invoke Querium ---PsQs15 [1] Math input error (Conditional) Premise [2] A Simplification [1] [3] Math input error Addition [2] [4] Math input error Conditional Discharge [1], [3]
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2.

We know that the show starts at 7 or 8 on some day. We want to show that if it is not true that the show starts on Saturday at 7 then, if it starts at Saturday at a time other than 8, they will burn down the theater.

Assign the following names to basic statements:

E : Show is E arly (at 7).

L : Show is L ate (at 8).

S : Show is on S aturday.

B : Theater B urns down.

Prove: E โˆจ L

  • (ยฌ( E โˆง S )) โ†’ (( S โˆง ยฌ L ) โ†’ B )

(Hint: Employ the Conditionalization rule more than once.)

Answer.
Invoke Querium. ---Theater questionId: Theater problemType: gradeLogicProof questionTitle: Using Conditionalization questionDisplayText: none problemGoal: (๏ƒ˜(E ๏ƒ™ S)) ๏‚ฎ ((S ๏ƒ™ ๏ƒ˜L) ๏‚ฎ B) initialProblemState: the one premise hints: Employ the Conditionalization rule more than once. *** [1] E ๏ƒš L Premise [2] ๏ƒ˜(E ๏ƒ™ S) (Conditional) Premise [3] S ๏ƒ™ ๏ƒ˜L (Conditional) Premise [4] S Simplification [3] [5] ๏ƒ˜E ๏ƒš ๏ƒ˜S De Morgan [2] [6] ๏ƒ˜E Disjunctive Syllogism [4], [5] [7] L Disjunctive Syllogism [1], [6] [8] ๏ƒ˜L Simplification [3] [9] B Contradictory Premises [7], [8] [10] (S ๏ƒ™ ๏ƒ˜L) ๏‚ฎ B Conditional Discharge [3], [9] [11] (๏ƒ˜(E ๏ƒ™ S)) ๏‚ฎ ((S ๏ƒ™ ๏ƒ˜L) ๏‚ฎ B) Conditional Discharge [2], [10]
Solution.
Explanation (to appear on new Quest slide after return from Querium): Notice the two uses of the Conditionalization rule (clearly marked with vertical lines so that weโ€™re careful to discharge their premises). They are what let us form the implications that we need. You can see that, by the time we assert our conclusion [11], we have discharged both conditional premises.
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3.

Either Joe or Mary or Sally will go to New York. If Paul stays home, then Joe will not go. Therefore, if Paul stays home and Mary does not go to New York, Sally must go to New York.

Assign the following names to basic statements:

J : Joe will go to New York.

M : Mary will go to New York.

S : Sally will go to New York.

P : Paul stays home.

Prove: J โˆจ ( M โˆจ S ) Joe or Mary or Sally will go to New York.

P โ†’ ยฌ J If Paul stays home, Joe will not go to New York.

โˆด ( P โˆง ยฌ M ) โ†’ S If Paul stays home and Mary doesnโ€™t go to NY, Sally must go to New York.

Answer.
Invoke Querium. ---NewYork questionId: NewYork problemType: gradeLogicProof questionTitle: Using Conditionalization questionDisplayText: none problemGoal: (P ๏ƒ™ ๏ƒ˜M) ๏‚ฎ S initialProblemState: the two premises hints: none [1] J ๏ƒš (M ๏ƒš S) Premise [2] P ๏‚ฎ ๏ƒ˜J Premise [3] P ๏ƒ™ ๏ƒ˜M Conditional Premise [4] P Simplification [3] [5] ๏ƒ˜J Modus Ponens [2], [4] [6] ๏ƒ˜M Simplification [3] [7] ๏ƒ˜J ๏ƒ™ ๏ƒ˜M Conjunction [5], [6] [8] ๏ƒ˜(J ๏ƒš M) De Morgan [7] [9] (J ๏ƒš M) ๏ƒš S Associativity of or [1] [10] S Disjunctive Syllogism [8], [9] [11] (P ๏ƒ™ ๏ƒ˜M) ๏‚ฎ S Conditional Discharge [3], [10]
Solution.
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4.

4. Prove that the following claim is a tautology (in other words, derive it without any premises):

(ยฌ p โˆง (ยฌ q โ†’ p )) โ†’ q

Answer.
---PsQs40 Invoke Querium
Solution.