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1. Let L = {s, t, u}. Use the rules of natural derivation to determine whether there is a u V t derived from s V u. Justify your answer is correct.

2. Let L = {p, q, r}. Take the sentence A = p → (q ↔ ¬r) as an example.

(a) Determine whether there is a Horn sentence in SL that is logically equivalent to A to justify your answer, fully stating any statement or result you rely on.

(b) Use your answer to part (a) to briefly describe an advantage or disadvantage of using Horn sentence logic instead of propositional logic reason. Logic as knowledge representation.

3. Suppose L = {p, q} and consider the following ranking model R = (V, ?), where V = {T T, T F, F T, F F} and ? are given in tabular form as follows: FT TF TT FF (Each estimate is represented as a pair representing the true value p, q, respectively, on the left, and the estimate appears in the table above, the more normal it is considered to be.) Explicitly state whether the following conditions hold in R. Justify your answer in each case. (a) p |∼R ¬q (b) q → p |∼R p