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

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.

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) : $$\exists x (W(x) \to S(x))$$.

• So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday? Jan 22, 2019 at 4:04