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