Kenneth Rosen Edition 6th Exercise 1.5 Question 9 a (Page No. 73)

Question

How do I proceed with the question below and then solve it?

For each sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises.

• "If I take the day off, it either rains or snows."
• "I took Tuesday off or I took Thursday off."
• "It was sunny on Tuesday."
• "It did not snow on Thursday."

Please help me in regard with this question. Thank You. 

Answer 1

Premises for this argument :

P1. I take Thursday off $\implies$ Rain(Thursday) $\vee$ Snow(Thursday)

P2. I take Tuesday off $\implies$ Rain(Tuesday) $\vee$ Snow(Tuesday)

P3. (I took Tuesday off) $\vee$ (I took Thursday off.)

P4. $\sim \left ( \text{Snow(Tuesday)} \vee \text{Rain(Tuesday)} \right )$

P5. $\sim$ Snow(Thursday)

Explanation of premises :

• P1 and P2 are one way implications.
• P3 is a disjunction proposition
• P4 was actually "Sunny on Tuesday". We have assumed No rain and No snow on tuesday. Using de morgan's, converted it to a negation of a disjunction.
• P5. Is a negation for Snow on Thursday

To get a conclusion Assume all premises are TRUE:

Now,

P4 and P2. apply Modus Tollens.  (contrapositive of P2 is true)

• => P6. I did't take Tuesday off.

P3 and P6. apply Disjunctive Syllogism (one of the proposition in P3 must be true to make P3 true)

• => P7. I took Thursday off.

P7 and P1. apply  Modus ponens ( P1 is one way implication, so, LHS true means RHS must be true to make P1 true )

• => P8. Snow(Thursday) $\vee$ Rain(Thursday)

P8 and P5. apply  Disjunctive Syllogism again. (one of the proposition in P8 must be true to make P8 true)

• => P9. Rain(Thursday) 

PS: P2 LHS becomes false. But it is an implicaiton, LHS false means proposition is true anyway, ok with our all premises TRUE assumption.

Comments

your proposition will be true (true means tuesday was sunny) even if : Snow(tuesday) = true and Rain(tuesday) = false.

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"

• P4 was actually "Sunny on Tuesday". We have assumed No rain and No snow on tuesday. Using de morgan's, converted it to a negation of a disjunction.

"

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Thanks for the clarification