Show the following formulas to be S5-Valid. 

Use the S5 rules given in chapter 2, i.e. Pr, MN, <>S5, []S5, []T.

a.        [][](p É p)

b.        <><>p É <>p

Translate the following arguments into S5 and test for validity. Create your own dictionary (ignoring tense changes and other grammatical irrelevancies.) 

Use the S5 rules given in chapter 2, i.e. Pr, MN, <>S5, []S5, []T.

a.        If I am to see the Olympics then I’ll need to travel to China ; but it’s not possible to travel to China without a visa, so I’ll need one of those.  Now it’s possible that I won’t get a visa, so it’s possible that I won’t see the Olympics.

b.        It is necessarily true that if I know that there will be a sea battle tomorrow, then there will be a sea battle tomorrow. Therefore if I know there will be a sea battle tomorrow, the sea battle tomorrow is necessary.

Show that each of the following formulas is valid in K.

a.        []p É (<>q É <>(p & q))

b.        <>(p É q) º ([]p É <>q)

Each of the following formulas is valid in at least one of T, S4, and S5. What is the weakest logic in which each is valid? Show this with a truth tree. (The truth tree is the important part.)

a.        [](p É []<>p)

b.        (<>~p v <>~q) v <>(p v q)

a.        Give a counterexample in S4 for [](p É []<>p)

b.        Give a counterexample in K for (<>~p v <>~q) v <>(p v q)