Explain the rules of inference used to obtain each conclusion from the premises (Rosen 8th Ed):

Question

I'm having a difficult time with the following - seems like a lot of work and I'm unsure if my conclusions makes sense when translated back to english.

1. If I work, it is either sunny or partly sunny (Premise): $\forall x (W(x) \to (S(x) \lor P(x))$
2. I worked last Monday or I worked last Friday (Premise): $W_{Monday} \lor W_{Friday}$
3. It was not sunny on Tuesday (Premise): $\lnot S_{Tuesday}$
4. It was not partly sunny on Friday (Premise): $\lnot P_{Friday}$

$5. W_{Friday} \to (S_{Friday} \lor P_{Friday})$ (Universal Instantiation 1.)

$6. P_{Friday} \lor (\lnot W_{Friday} \lor S_{Friday} )$ (Logical Equivalence, Commutative & Associative Laws)

$7.\lnot W_{Friday} \lor S_{Friday}$ (Disjunctive Syllogism 4. & 6.)

$8.W_{Monday} \lor S_{Friday}$ (Resolution 2. & 7.)

$9. W_{Monday} \to (S_{Monday} \lor P_{Monday})$ (Universial Instantiation 1.)

$10. \lnot W_{Monday} \lor (S_{Monday} \lor P_{Monday})$ (Logical Equivalence)

$11.S_{Friday} \lor (S_{Monday} \lor P_{Monday}) \equiv \lnot S_{Friday} \to (S_{Monday} \lor P_{Monday})$ (Resolution 8. & 10.)

$12. \exists x \exists y(\lnot S(x) \to (S(y) \lor P(y))$ (Existential Generalization 11. with 3.) Conclusion: There exists a day when it was not sunny, while on another day it was either sunny or partly sunny.

Seems rather inconclusive.

Answer 1

Hint

The premises refer to three days : Mon, Tue, Fri.

Thus, it can be useful to use all of them in Universal instantiation of 1) to get :

5) $W(F) → (S(F) ∨ P(F))$ i.e. $\lnot W(F) ∨ (S(F) ∨ P(F))$

6) $\lnot W(T) ∨ (S(T) ∨ P(T))$

and

7) $\lnot W(M) ∨ (S(M) ∨ P(M))$.

Then, using Resolution, we get :

8) $\lnot W(T) ∨ P(T)$ --- from 3) and 6)

9) $\lnot W(F) ∨ S(F)$ --- from 4) and 5)

10) $W(F) \lor S(M) ∨ P(M)$ --- from 2) and 7)

11) $W(M) \lor S(F)$ --- from 2) and 9).

All this is not very useful...

8) is $W(T) \to P(T)$ and from it : $\exists x (W(x) \to P(x))$.

In the same way, from 9) : $\exists x (W(x) \to S(x))$.