logic pset11

A. Translation

Translate the following sentences into predicate logic notation. Restrict yourself to the following vocabulary (and the equality symbol):

    Pxy = x is a parent of y
    Lxy = x loves y
    a   = Alice
    b   = Bob

Assume that being a child means having a parent, being siblings means having at least one parent in common, and being cousins means having at least one grandparent in common. Assume also that quantifiers apply only to people.

Everyone except Alice has a sibling.
Alice is Bob’s cousin.
Everybody loves their own grandchildren.
No person loves children unless they are a parent.

B. Proofs and Counterexamples

Consider the following two sentences:
1. ∀xFx→P
2. ∀x(Fx→P)
One of these two implies the other, but not vice versa. Give a proof for the implication, and give a counterexample for the non-implication.
Consider the following two sentences:
1. ∀x∀y∃z(Rxz∧Ryz)
2. ∃z∀xRxz
One of these two implies the other, but not vice versa. Give a proof for the implication, and give a counterexample for the non-implication.
Could there be a correctly written proof with the following line fragments? Explain your answer with reference to soundness or completeness.
```
1    (1) ∃x(Fx∧Gx)     A
2    (2) ∃x(Gx∧Hx)     A

1,2  (n) ∃x(Fx∧Hx)     
```
Could there be a correctly written proof with the following line fragments? Explain your answer with reference to soundness or completeness. (Hint: ∃x∀y(Fx→Fy) is logically equivalent to ∃x(Fx→∀yFy).)
```
1   (1) ¬∃x∀y(Fx→Fy)

1   (n) ∀x(Fx↔¬Fx)
```